UNIT 6
Exponential Relationships CONTENTS COMMON CORE
F-LE.A.2 F-BF.A.1a F-LE.A.2 F-IF.C.7e F-BF.B.3
COMMON CORE
A-CED.A.1 F-IF.C.7e S-ID.B.6a F-LE.A.1c
633A
Unit 6
MODULE 13
Geometric Sequences and Exponential Functions
Lesson 14.1 Lesson 14.2 Lesson 14.3 Lesson 14.4 Lesson 14.5
Understanding Geometric Sequences Constructing Geometric Sequences. . Constructing Exponential Functions . Graphing Exponential Functions. . . . Transforming Exponential Functions .
MODULE 15
Exponential Equations and Models
Lesson 15.1
Using Graphs and Properties to Solve Equations with Exponents . . . . . . . . . . . Modeling Exponential Growth and Decay . . Using Exponential Regression Models . . . . Comparing Linear and Exponential Models
Lesson 15.2 Lesson 15.3 Lesson 15.4
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UNIT 6
Unit Pacing Guide 45-Minute Classes Module 14 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 14.1
Lesson 14.2
Lesson 14.2
Lesson 14.3
Lesson 14.4
DAY 6
DAY 7
DAY 8
Lesson 14.4
Lesson 14.5
Module Review and Assessment Readiness
Module 15 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 15.1
Lesson 15.1
Lesson 15.2
Lesson 15.2
Lesson 15.3
DAY 6
DAY 7
DAY 8
Lesson 15.4
Module Review and Assessment Readiness
Unit Review and Assessment Readiness
DAY 1
DAY 2
DAY 3
DAY 4
Lesson 14.1 Lesson 14.2
Lesson 14.2 Lesson 14.3
Lesson 14.4
Lesson 14.5 Module Review and Assessment Readiness
DAY 1
DAY 2
DAY 3
DAY 4
Lesson 15.1
Lesson 15.2
Lesson 15.3 Lesson 15.4
Module Review and Assessment Readiness Unit Review and Assessment Readiness
90-Minute Classes Module 14
Module 15
Unit 6
633B
Program Resources PLAN
ENGAGE AND EXPLORE
HMH Teacher App Access a full suite of teacher resources online and offline on a variety of devices. Plan present, and manage classes, assignments, and activities.
Real-World Videos Engage students with interesting and relevant applications of the mathematical content of each module.
Explore Activities Students interactively explore new concepts using a variety of tools and approaches.
ePlanner Easily plan your classes, create and view assignments, and access all program resources with your online, customizable planning tool.
Professional Development Videos Authors Juli Dixon and Matt Larson model successful teaching practices and strategies in actual classroom settings. QR Codes Scan with your smart phone to jump directly from your print book to online videos and other resources. DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A
Teacher’s Edition Support students with point-of-use Questioning Strategies, teaching tips, resources for differentiated instruction, additional activities, and more. DONOT NOTEDIT--Changes EDIT--Changesmust mustbe bemade madethrough through"File "Fileinfo" info" DO CorrectionKey=NL-A;CA-A CorrectionKey=NL-A;CA-A
Name Name
Isosceles and Equilateral Triangles
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Class Class
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Common Core Math Standards
Investigating Isosceles Triangles INTEGRATE TECHNOLOGY
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G-CO.C.10
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CONNECT VOCABULARY Ask a volunteer to define isosceles triangle and have students give real-world examples of them. If possible, show the class a baseball pennant or other flag in the shape of an isosceles triangle. Tell students they will be proving theorems about isosceles triangles and investigating their properties in this lesson.
Class
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Proving the Isosceles Triangle Theorem and Its Converse
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© Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company
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© Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company
PREVIEW: LESSON PERFORMANCE TASK
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Language Objective
Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles?
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An isosceles triangle is a triangle with at least two congruent sides.
Students have the option of completing the isosceles triangle activity either in the book or online.
Resource Locker
G-CO.C.10 Prove theorems about triangles.
Explore
CC
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COMMON CORE
EXPLORE
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22.2
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DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A
LESSON
Name
Base Base angles
PROFESSIONAL DEVELOPMENT
TEACH
ASSESSMENT AND INTERVENTION
Math On the Spot video tutorials, featuring program authors Dr. Edward Burger and Martha Sandoval-Martinez, accompany every example in the textbook and give students step-by-step instructions and explanations of key math concepts.
Interactive Teacher Edition Customize and present course materials with collaborative activities and integrated formative assessment.
C1
Lesson 19.2 Precision and Accuracy
Evaluate
1
Lesson XX.X ComparingLesson Linear, Exponential, and Quadratic Models 19.2 Precision and Accuracy
teacher Support
1
EXPLAIN Concept 1
Explain
The Personal Math Trainer provides online practice, homework, assessments, and intervention. Monitor student progress through reports and alerts. Create and customize assignments aligned to specific lessons or Common Core standards. • Practice – With dynamic items and assignments, students get unlimited practice on key concepts supported by guided examples, step-by-step solutions, and video tutorials. • Assessments – Choose from course assignments or customize your own based on course content, Common Core standards, difficulty levels, and more. • Homework – Students can complete online homework with a wide variety of problem types, including the ability to enter expressions, equations, and graphs. Let the system automatically grade homework, so you can focus where your students need help the most! • Intervention – Let the Personal Math Trainer automatically prescribe a targeted, personalized intervention path for your students. 2
3
4
Question 3 of 17
Concept 2
Determining Precision
ComPLEtINg thE SquArE wIth EXPrESSIoNS Avoid Common Errors Some students may not pay attention to whether b is positive or negative, since c is positive regardless of the sign of b. Have student change the sign of b in some problems and compare the factored forms of both expressions. questioning Strategies In a perfect square trinomial, is the last term always positive? Explain. es, a perfect square trinomial can be either (a + b)2 or (a – b)2 which can be factored as (a + b)2 = a 2 + 2ab = b 2 and (a – b)2 = a 2 + 2ab = b 2. In both cases the last term is positive. reflect 3. The sign of b has no effect on the sign of c because c = ( b __ 2 ) 2 and a nonzero number squared is always positive. Thus, c is always positive. c = ( b __ 2 ) 2 and a nonzero number c = ( b __ 2 ) 2 and a nonzero number
5
6
7
View Step by Step
8
9
10
11 - 17
Video Tutor
Personal Math Trainer
Textbook
X2 Animated Math
Solve the quadratic equation by factoring. 7x + 44x = 7x − 10
As you have seen, measurements are given to a certain precision. Therefore,
x=
the value reported does not necessarily represent the actual value of the measurement. For example, a measurement of 5 centimeters, which is
,
Check
given to the nearest whole unit, can actually range from 0.5 units below the reported value, 4.5 centimeters, up to, but not including, 0.5 units above it, 5.5 centimeters. The actual length, l, is within a range of possible values:
Save & Close
centimeters. Similarly, a length given to the nearest tenth can actually range from 0.05 units below the reported value up to, but not including, 0.05 units above it. So a length reported as 4.5 cm could actually be as low as 4.45 cm or as high as nearly 4.55 cm.
?
!
Turn It In
Elaborate
Look Back
Focus on Higher Order Thinking Raise the bar with homework and practice that incorporates higher-order thinking and mathematical practices in every lesson.
Differentiated Instruction Resources Support all learners with Differentiated Instruction Resources, including • Leveled Practice and Problem Solving • Reading Strategies • Success for English Learners • Challenge Calculate the minimum and maximum possible areas. Round your answer to
Assessment Readiness
the nearest square centimeters.
The width and length of a rectangle are 8 cm and 19.5 cm, respectively.
Prepare students for success on high stakes tests for Integrated Mathematics 1 with practice at every module and unit
Find the range of values for the actual length and width of the rectangle.
Minimum width =
7.5
cm and maximum width <
8.5 cm
My answer
Assessment Resources
Find the range of values for the actual length and width of the rectangle.
Minimum length =
19.45
cm and maximum length < 19.55
Name ________________________________________ Date __________________ Class __________________ LESSON
1-1
cm
Name ____________ __________________ __________ Date __________________ LESSON Class ____________ ______
Precision and Significant Digits
6-1
Success for English Learners
Linear Functions
Reteach
The graph of a linear The precision of a measurement is determined bythe therange smallest unit or Find of values for the actual length and width of the rectangle. function is a straig ht line. fraction of a unit used. Ax + By + C = 0 is the standard form for the equat ion of a linear functi • A, B, and C are on. Problem 1 Minimum Area = Minimum width × Minimum length real numbers. A and B are not both zero. • The variables x and y Choose the more precise measurement. = 7.5 cm × 19.45 cm have exponents of 1 are not multiplied together are not in denom 42.3 g is to the 42.27 g is to the inators, exponents or radical signs. nearest tenth. nearest Examples These are NOT hundredth. linear functions: 2+4=6 no variable x2 = 9 exponent on x ≥ 1 xy = 8 x and y multiplied 42.3 g or 42.27 g together 6 =3 Because a hundredth of a gram is smaller than a tenth of a gram, 42.27 g x in denominator x is more precise. 2y = 8 y in exponent Problem 2 In the above exercise, the location of the uncertainty in the linear y = 5 y in a square root measurements results in different amounts of uncertainty in the calculated Choose the more precise measurement: 36 inches or 3 feet. measurement. Explain how to fix this problem. Tell whether each function is linear or not. 1. 14 = 2 x 2. 3xy = 27 3. 14 = 28 4. 6x 2 = 12 x ____________
Reflect
____
________________
_______________
The graph of y = C is always a horiz ontal line. The graph always a vertical line. of x = C
_______________
is
Unit 6
Send to Notebook
_________________________________________________________________________________________
2. An object is weighed on three different scales. The results are shown Explore in the table. Which scale is the most precise? Explain your answer. Measurement
____________________________________________________________
• Tier 1, Tier 2, and Tier 3 Resources
Examples
1. When deciding which measurement is more precise, what should you Formula consider?
Scale
Tailor assessments and response to intervention to meet the needs of all your classes and students, including • Leveled Module Quizzes • Leveled Unit Tests • Unit Performance Tasks • Placement, Diagnostic, and Quarterly Benchmark Tests
Your Turn
y=1 T
x=2
y = −3
x=3
633D
Math Background Geometric Sequences
COMMON CORE
F-LE.A.2
LESSON 14.1 Informally, we move from one term of an arithmetic sequence to the next by addition. In a geometric sequence, we move from one term to the next by multiplication. More precisely, in a geometric sequence, the ratio of each term to the preceding term is constant. This constant is called the common ratio. The sequence 1, 3, 9, 27, 81, . . . is an example of a geometric sequence. The common ratio is 3. Each term of the sequence is multiplied by 3 to get the next term.
Exponential Functions
COMMON CORE
F-IF.C.7e
LESSONS 14.3 and 14.4 To understand the connection between geometric sequences and exponential functions, we begin with the definition of exponential function: a function of the form y = ab x, where a and b are real numbers, a ≠ 0, b > 0, and b ≠ 1. In this course, students have seen b x defined for limited x-values only (namely, rational numbers), but the domain of an exponential function can be the set of all real numbers. Students will fill in this gap in future courses and can be assured in the meantime that they may draw a smooth, continuous curve when graphing an exponential function. For a > 0, the range of an exponential function is the set of all positive real numbers, as illustrated in the x graph of y = 3(2) .
16
y (2, 12)
12 8 y = 3(2)x 4 -4
633E
Unit 6
-2
0
(1, 6) (0, 3) 2
x 4
When the domain is restricted to whole numbers, the graph becomes a set of discrete points, the first few of which are shown above. The points’ y-values are 3, 6, 12, ... ; this is n–1 precisely the geometric sequence defined by a n = 3(2) . In other words, a geometric sequence is simply an exponential function with a restricted domain. (Note that the domain is usually restricted to the set of whole numbers greater than 0, so that the first term of the sequence corresponds to the input value 1 rather than to 0.)
Exponential Growth and Decay
COMMON CORE
F-LE.A.1c
LESSON 15.2 Exponential growth and decay are important applications of exponential functions. Exponential growth occurs when a quantity increases by the same rate in each time period. Thus, exponential growth can be understood as an extension of percent change in which a percent increase is repeatedly applied. Exponential decay occurs when a quantity decreases by the same rate in each time period and, like exponential growth, can be understood as an extension of percent change in which a percent decrease is repeatedly applied. In an exponential growth situation with initial amount a (a > 0) and rate of growth after one time period, r, expressed as a decimal, the new amount is a + ar, or a(1 + r). After the second time period, the new amount is 2 a(1 + r)(1 + r), or a(1 + r) . Continuing in this way shows that after t time periods, the final amount y is given by t y = a(1 + r) . For exponential decay, a similar argument shows that the final amount y after t time periods is given t by y = a(1 - r) . An important attribute of exponential growth and decay is the fact that the amount added or subtracted in each time period is proportional to the amount already present. For exponential growth, this means that as the amount becomes greater, the amount of increase in each time period also becomes greater. Contrast this to linear growth, in which the amount of increase remains constant.
PROFESSIONAL DEVELOPMENT
COMMON CORE
S-ID.B.6a
LESSON 15.3 Exponential regression is the process of finding an exponential function that approximates the relationship between two variables in a data set. The original data, along with the best-fit curve given by the exponential regression process, can be represented on an scatter plot. The correlation coefficient (r) indicates how well the regression model fits the data. The exponential function y = f(x) = ab x can be used to model the data set. An exponential regression is used when a straight line does not fit the data, but an exponential function might. A residual is the difference between the observed y-value in the data set and the predicted y-value (y d – y m). Residuals can be used to assess how well a model fits a data set with the following guidelines: • The numbers of positive and negative residuals are roughly equal. • The residuals are randomly distributed about the x-axis, not in a pattern. • The absolute value of the residuals is small relative to the data.
Linear and Exponential Models
COMMON CORE
S-ID.B.6a
LESSON 15.4 A linear model should be used when the amount of increase or decrease in each interval is constant, such as a fixed dollar increase. An exponential model is appropriate when the increase or decrease per interval grows, such as a fixed percent increase. Consider the following situation. Two students each have $20 in savings. Student 1 increases how much he has saved by $4 per month by saving $4 per month. Student 2 increases how much he has saved by 1.5% per month. Graphing the above scenario shows the number of weeks it takes for Student 2 to have more money saved. Students’ Savings
Money ($)
Exponential Regression
70 60 50 40 30 20 10 0
Student 1 Student 2
1 2 3 4 5 6 7 8 9 Weeks
Unit 6
633F
UNIT
6
UNIT 6
Exponential Relationships
MODULE
Exponential Relationships
MATH IN CAREERS Unit Activity Preview
14
Geometric Sequences and Exponential Functions MODULE
15
Exponential Equations and Models
After completing this unit, students will complete a Math in Careers task by writing and interpreting exponential functions based on a graph. Critical skills include modeling real-world situations and interpreting functional relationships.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Dragon Images/Shutterstock
For more information about careers in mathematics as well as various mathematics appreciation topics, visit The American Mathematical Society at http://www.ams.org.
MATH IN CAREERS
Financial Research Analyst Financial research analysts perform quantitative analysis of market conditions. They use statistics and mathematical models to determine investment strategies and communicate findings. If you are interested in a career as a financial research analyst, you should study these mathematical subjects: • Algebra • Business math • Statistics • Calculus • Differential equations
Unit 6
633
Research other careers that require understanding how to calculate interest rates. Check out the career activity at the end of the unit to find out how financial research analysts use math.
TRACKING YOUR LEARNING PROGRESSION
IN1_MNLESE389762_U6UO.indd 633
4/19/14 10:53 PM
Before
In this Unit
After
Students understand: • systems of linear equations • systems of linear inequalities
Students will learn about: • geometric sequences • exponential functions
Students will study: • adding polynomial expressions • subtracting polynomial expressions • multiplying polynomial expressions • special products of binomials
633
Unit 6
Reading Start -Up
Reading Start Up
Vocabulary Review Words ✔ explicit rule (fórmula explícita) exponent (exponente) ✔ linear function (función lineal) ✔ recursive rule (fórmula recurrente) ✔ sequence (sucesión) ✔ term (término)
Visualize Vocabulary Use the ✔ words to complete the Summary Triangle. Write one word in each box.
A
sequence is a list of numbers in a specific order.
Have students complete the activities on this page by working alone or with others.
VISUALIZE VOCABULARY The summary triangle graphic helps students review vocabulary associated with sequences. If time allows, brainstorm other mathematical relationships among the words.
Preview Words exponential decay (decremento exponencial) exponential function (función exponencial) exponential growth (crecimiento exponencial)
Each element is called a term and has a position number.
UNDERSTAND VOCABULARY Use the following explanations to help students learn the preview words.
An explicit rule defines the nth term as a function of n and can be used to find any specific term without finding any of the previous terms.
In an exponential function the independent variable is an exponent. If the function increases as the independent variable increases, it models exponential growth. If the function decreases as the independent variable increases, it models exponential decay.
A recursive rule defines the nth term by relating it to one or more previous terms. It cannot directly be used to find a specific term.
To become familiar with some of the vocabulary terms in this unit, consider the following. You may refer to the module, the glossary, or a dictionary. 1.
A function of the form y = a(b) is called an x
When b > 1, the function represents
exponential function
.
exponential growth
When 0 < b < 1, the function represents
. exponential decay
.
Active Reading
© Houghton Mifflin Harcourt Publishing Company
Understand Vocabulary
ACTIVE READING Students can use these reading and note-taking strategies to help them organize and understand the new concepts and vocabulary. Encourage them to use mathematical vocabulary precisely and to question terminology that is unclear or misleading. Have students include any additional vocabulary words that they feel will be helpful in their key-term fold.
Key-Term Fold Before beginning the unit, create a key-term fold note to help you organize what you learn. Write a vocabulary term on each tab of the key-term fold. Under each tab, write the definition of the term and an example of the term.
Unit 6
ADDITIONAL RESOURCES
634
Differentiated Instruction IN1_MNLESE389762_U6UO.indd 634
4/19/14 9:19 AM
• Reading Strategies
Unit 6
634
MODULE
14
14 MODULE
Geometric Sequences and Exponential Functions
Geometric Sequences and Exponential Functions
Essential Question: How can you use geometric
sequences and exponential functions to solve real-world problems?
ESSENTIAL QUESTION:
LESSON 14.1
Understanding Geometric Sequences LESSON 14.2
Constructing Geometric Sequences
Answer: Geometric sequences and exponential functions have a wide range of applications in fields including economics and biology.
LESSON 14.3
Constructing Exponential Functions LESSON 14.4
This version is for PROFESSIONAL DEVELOPMENT Algebra 1 and Geometry only VIDEO
Graphing Exponential Functions LESSON 14.5
Transforming Exponential Functions
Professional Development Video
Professional Development my.hrw.com
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©RLHambley/Shutterstock
Author Juli Dixon models successful teaching practices in an actual high-school classroom.
REAL WORLD VIDEO Pythons originally kept as pets but later released into the Florida ecosystem find themselves in an environment with no natural predators and prey ill-equipped to evade or defend itself. As a result, the python population can grow exponentially, causing havoc among local wildlife and pets.
MODULE PERFORMANCE TASK PREVIEW
What Does It Take to Go Viral? You have just created a great video, and you share it with some of your friends. Then each of them shares it with the same number of their own friends. If this pattern continues, to how many friends should you show your video to make it go viral within a few days? Let’s find out!
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Access a full suite of teaching resources when and where you need them: • Access content online or offline • Customize lessons to share with your class • Communicate with your students in real-time • View student grades and data instantly to target your instruction where it is needed most
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PERSONAL MATH TRAINER Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, Common Core-aligned practice tests.
4/19/14 10:54 AM
Are YOU Ready?
Are You Ready?
Complete these exercises to review skills you will need for this chapter.
Exponents
ASSESS READINESS
Evaluate (-6) . 3
Example 1
Write the base -6 multiplied by itself 3 times.
(-6)3 = (-6)(-6)(-6)
(-6)(-6)(-6) = -216
Multiply.
Use the assessment on this page to determine if students need strategic or intensive intervention for the module’s prerequisite skills.
• Online Homework • Hints and Help • Extra Practice
Evaluate each power. 2. (-3)
1. 2 4
16 Simplify x 3 · x 5.
Example 2
x3 · x5 = x3 + 5 = x8
5
3. 5 0
-243
1
ASSESSMENT AND INTERVENTION
When multiplying numbers with the same base, add the exponents.
Simplify. 4. x · x 6
5. x 3y 2 · y 4
x7
x 3y 6 3 2
7. 4mno · 7n 2o 2 · mn
6. 3a 2b · 5a 2b 4
1
Personal Math Trainer will automatically create a standards-based, personalized intervention assignment for your students, targeting each student’s individual needs!
28m 2n 4o 3
15a 4b 5
Algebraic Expressions Substitute 6 for x.
© Houghton Mifflin Harcourt Publishing Company
2 2 Evaluate _ x for x = 6. 3 2 2 (6) _ 3 2 (36) _ 3 24
Example 3
Evaluate the power. Multiply.
Evaluate each expression for the given value of the variables. 1 8. _x 3 for x = 4 2
1 2
10. 8x 2 for x = _
2
3 x 4 for x = -2 9. __ 4
12
32
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Tier 1 Lesson Intervention Worksheets Reteach 14.1 Reteach 14.2 Reteach 14.3 Reteach 14.4 Reteach 14.5
1 3
11. 18x 3 for x = -_
2 -__ 3
TIER 1, TIER 2, TIER 3 SKILLS
ADDITIONAL RESOURCES See the table below for a full list of intervention resources available for this module. Response to Intervention Resources also includes: • Tier 2 Skill Pre-Tests for each Module • Tier 2 Skill Post-Tests for each skill
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Response to Intervention Tier 2 Strategic Intervention Skills Intervention Worksheets 2 Algebraic Expressions 5 Exponents 11 Multi-Step Equations
Differentiated Instruction
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Tier 3 Intensive Intervention Worksheets available online Building Block Skills 19, 22, 23, 24, 27, 29, 30, 40, 59, 69, 76, 81, 98, 100
Challenge worksheets Extend the Math Lesson Activities in TE
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LESSON
14.1
Name
Understanding Geometric Sequences
Class
Date
14.1 Understanding Geometric Sequences Essential Question: How are the terms of a geometric sequence related? Resource Locker
Common Core Math Standards The student is expected to: COMMON CORE
Explore 1
F-LE.A.2
Construct… geometric... sequences, given a graph, a description of a relationship, or two input-output pairs… Also F-LE.A.3
The sequence 3, 6, 12, 24, 48, … is a geometric sequence. In a geometric sequence, the ratio of successive terms is constant. The constant ratio is called the common ratio, often represented by r.
Mathematical Practices COMMON CORE
A
Complete each division.
MP.7 Using Structure
6= 2 _ 3
Language Objective Explain to a partner how to tell whether a sequence is a geometric sequence.
C
Use the common ratio you found to identify the next term in the geometric sequence.
2 = 96 .
Reflect
© Houghton Mifflin Harcourt Publishing Company
View the Engage section online. Discuss the photo and examples of payment plans students might use when charging for odd jobs. Then preview the Lesson Performance Task.
48 = 2 _ 24
24 = 2 _ 12
The common ratio r for the sequence is 2 .
The next term is 48 ·
Essential Question: How area the terms of a geometric sequence related?
PREVIEW: LESSON PERFORMANCE TASK
12 = 2 _ 6
B
ENGAGE The terms of a geometric sequence are related by a common ratio, often represented by r.
Exploring Growth Patterns of Geometric Sequences
1.
Suppose you know the twelfth term in a geometric sequence. What do you need to know to find the thirteenth term? How would you use that information to find the thirteenth term? You need to know the common ratio, r. You can multiply the twelfth term by r.
2.
Discussion Suppose you know only that 8 and 128 are terms of a geometric sequence. Can you find the term that follows 128? If so, what is it? Only if you know that 8 and128 are successive terms. In that case, the common ratio is 32, and the next term is 4096. However, 8 and 128 could be terms of a different geometric sequence. For example, in the geometric sequence 8, 16, 32, 64, 128, ..., the next term is 256.
Explore 2
Comparing Growth Patterns of Arithmetic and Geometric Sequences
Recall that in arithmetic sequences, successive terms differ by the same nonzero number d, called the common difference. In geometric sequences, the ratio r of successive terms is constant. In this Explore, you will examine how the growth patterns in arithmetic and geometric sequences compare. In particular, you will look at the arithmetic sequence 3, 5, 7, ... and the geometric sequence 3, 6, 12, ... . Module 14
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The tables shows the two sequences.
3, 5, 7, ... Term Number 1 2 3 4 5
EXPLORE 1
3, 6, 12, ... Term 3 5 7 9 11
Term Number 1 2 3 4 5
Term 3 6 12 24 48
Exploring Growth Patterns of Geometric Sequences
A
The common difference d of the arithmetic sequence is 5 - 3 = 2. The common ratio r of the geometric 6 = sequence is _ 2 . 3
B
Complete the table. Find the differences of successive terms.
INTEGRATE TECHNOLOGY Have students complete the Explore activity in either the book or online lesson.
Arithmetic: 3, 5, 7, ... Term Difference
CONNECT VOCABULARY
Term Number 1
3
—
2
5
5-3=
3
7
7-5= 2
4
9
9-7=
5
11
11 - 9 = 2
Make sure that students understand the meanings of successive terms and ratio of successive terms. You can explain that two successive terms are two terms that are next to each other in the sequence. Have students give examples of pairs of successive terms. Explain that successive terms can also be called consecutive terms.
2
2
Geometric: 3, 6, 12, ... Term Difference
Term Number
3
—
2
6
6-3=
3
12
12 - 6 = 6
4
24
24 - 12 = 12
5
48
48 - 24 = 24
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EXPLORE 2
© Houghton Mifflin Harcourt Publishing Company
1
Comparing Growth Patterns of Arithmetic and Geometric Sequences QUESTIONING STRATEGIES How are the graphs of geometric sequences and arithmetic sequences alike? How are they different? Possible answer: They can both be represented by a function with a domain that is the set of positive integers, or a subset of consecutive positive integers beginning with 1. The graph of a geometric sequence follows a curve, while the graph of an arithmetic sequence is linear.
Lesson 1
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Learning Progressions
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In an earlier module, students studied arithmetic sequences and wrote general recursive and explicit rules for them. Students used these rules to solve real-world problems involving arithmetic sequences. In this module, students will learn about geometric sequences and exponential functions. In the next module, students will learn more about exponential functions, including exponential growth and decay functions.
Understanding Geometric Sequences
638
C
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Show students a graph of the first 4 terms of
Compare the growth patterns of the sequences based on the tables.
For the arithmetic sequence, the differences are equal. The terms of the arithmetic sequence increase by a fixed amount. For the geometric sequence, the differences increase. The terms of the geometric sequence increase by an increasing amount.
f(n) = 2 + 2(n - 1) and f(n) = 2 · 2n - 1. Point out that the shape formed by the points indicates whether the graph represents an arithmetic or geometric sequence. Look at three or more points before determining whether the graph is linear or exponential, since arithmetic and geometric sequences can have the same first two terms.
D
Graph both sequences in the same coordinate plane. Compare the growth patterns based on the graphs. Sequence Patterns 50
y
Rope B (ft)
40 30
Geometric sequence
20 10
Arithmetic sequence x
EXPLAIN 1
0
1
2
3
4
5
Rope A (ft)
Extending Geometric Sequences
For the arithmetic sequence, the slope is constant, so the vertical distances between successive points are the same. For the geometric sequence, the vertical distances
AVOID COMMON ERRORS
Reflect
3. © Houghton Mifflin Harcourt Publishing Company
When finding a common ratio, students might divide in the wrong order. Tell them they are really finding what each term is being multiplied by to get the next term, so the inverse operation (division), will produce r. For example, when finding the common 1 , ..., ratio of the geometric sequence 16, 4, 1, _ 4 divide 4 by 16.
between the points increase by increasing amounts.
Which grows more quickly, the arithmetic sequence or the geometric sequence? The geometric sequence
Explain 1
Extending Geometric Sequences
In Explore 1, you saw that each term of a geometric sequence is the product of the preceding term and the common ratio. Given terms of a geometric sequence, you can use this relationship to write additional terms of the sequence.
Finding a Term of a Geometric Sequence For n ≥ 2, the nth term, ƒ(n), of a geometric sequence with common ratio r is ƒ(n) = ƒ(n - 1)r.
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Peer-to-Peer Activity Have students work in pairs. Have one student write and graph a geometric sequence for which the common ratio r is greater than 1. Have the other student write and graph a geometric sequence for which the common ratio r is 0 < r < 1. Have the students compare the graphs. Have the students switch roles and repeat the exercise.
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Example 1
Find the common ratio r for each geometric sequence and use r to find the next three terms.
QUESTIONING STRATEGIES In giving the formula for the nth term of a geometric sequence, why does the book say “For n ≥ 2”? It is understood that n is an integer. The first term of the sequence is f (1). For n = 0 and 1, f(n - 1) is not defined.
6, 12, 24, 48, … 12 = 2, so the common ratio r is 2. _ 6 For this sequence, ƒ(1) = 6, ƒ(2) = 12, ƒ(3) = 24, and ƒ(4) = 48. ƒ(4) = 48, so ƒ(5) = 48(2) = 96. ƒ(5) = 96, so ƒ(6) = 96(2) = 192. ƒ(6) = 192, so ƒ(7) = 192(2) = 384. The next three terms of the sequence are 96, 192, and 384.
100, 50, 25, 12.5, … 50 = 0.5 , so the common ratio r is _ 100
0.5 .
For this sequence, ƒ(1) = 100, ƒ(2) = 50, ƒ(3) = 25, and ƒ(4) = 12.5. ƒ(4) = 12.5, so ƒ(5) = 12.5 (0.5) = 6.25 . ƒ(5) = 6.25 , so ƒ(6) = 6.25 (0.5) = ƒ(6) =
3.125 , so ƒ(7) =
3.125 .
3.125 (0.5) = 1.5625 .
The next three terms of the sequence are 6.25 ,
3.125 , and 1.5625 .
Reflect
Communicate Mathematical Ideas A geometric sequence has a common ratio of 3. The 4th term is 54. What is the 5th term? What is the 3rd term? The 5th term is 3 times the 4th, or 162. The 4th term, 54, is 3 times the 3rd term, so the 3rd term is 18.
Your Turn
Find the common ratio r for each geometric sequence and use r to find the next three terms. 5.
5, 20, 80, 320, ...
6.
20 _ =4=r 5 f(4) = 320, so f(5) = 320(4) = 1280.
f(5) = 1280, so f(6) = 1280(4) = 5120.
1 , ... 36, -12, 4, -_ 3 -12 1 =- =r 36 3 1 f(4) = - 1 , so f(5) = - 1 - 1 = . 9 3 3 3
_
f(5) =
f(6) = 5120, so f(7) = 5120(4) = 20,480. The next three terms are 1280, 5120, and 20,480.
_ _
_( _) _
1 . _1 , so f(6) = _1 (-_1 ) = -_ 9
_
9
3
27
1 _( _ ) = _ .
f(6) = - 1 , so f(7) = - 1 - 1 27 27 3
81
1 1 1 , and _ The next three terms are _, -_ . 9
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Lesson 1
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Multiple Representations
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Help students making connections between a table representing an arithmetic sequence and a graph of the sequence. For example, start with the table and have students make the graph. Then start with the graph and have students make the table. Do the same for a geometric sequence. Have students compare the graphs of the two sequences.
Understanding Geometric Sequences
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Explain 2
EXPLAIN 2
Recognizing Growth Patterns of Geometric Sequences in Context
You can find a term of a sequence by repeatedly multiplying the first term by the common ratio.
Recognizing Growth Patterns of Geometric Sequences in Context
Example 2
QUESTIONING STRATEGIES If you know that a sequence is a geometric sequence, how can you find the common ratio? Choose any term after the first term and divide it by the preceding term. The result is the common ratio.
A bungee jumper jumps from a bridge. The table shows the bungee’s jumpers height above the ground at the top of each bounce. The heights form a geometric sequence. What is the bungee jumper’s height at the top of the 5th bounce?
Bounce
Height (feet)
1
200
2
80
3
32
First bounce 200 ft
Find r. Second bounce 80 ft
80 = 0.4 = r _ 200
Third bounce 32 ft
ƒ(1) = 200 ƒ(2) = 80 = 200(0.4) or 200(0.4)
1
ƒ(3) = 32 = 80(0.4) = 200(0.4)(0.4) 2 = 200(0.4)
© Houghton Mifflin Harcourt Publishing Company
In each case, to get ƒ(n), you multiply 200 by the common ratio, 0.4, n -1 times. That is, you multiply n-1 200 by (0.4) . The jumper’s height on the 5th bounce is ƒ(5). 4 5- 1 Multiply 200 by (0.4) = (0.4) . 4 200(0.4) = 200(0.0256)
= 5.12 The height of the jumper on the 5th bounce is 5.12 feet.
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Connect Vocabulary Remind students that in a geometric sequence, the ratio of successive terms is constant. This constant ratio is called the common ratio, often written as r. Point out that it is called a common ratio because it is shared by all the pairs of successive terms. Discuss other uses of this meaning of the word common; for example, a common boys’ name.
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Example 2
HOME CONNECTION
A ball is dropped from a height of 144 inches. Its height on the 1st bounce is 72 inches. On the 2nd and 3rd bounces, the height of the ball is 36 inches and 18 inches, respectively. The heights form a geometric sequence. What is the height of the ball on the 6th bounce to the nearest tenth of an inch?
Have students think of a real-world situation that can be represented by a geometric sequence.
Find r. 36 = _ 1 =r _ 72 2 ƒ(1) = 72 ƒ(2) = 36
__ ) ( __ ) = 72( or 72 1 2
1 2
1
ƒ(3) = 18
( __ ) __ )( __ ) = 72( __ ) = 72( = 36
1 2 1 2 1 2
1 2
2
In each case, to get f(n), you multiply 72 by the common ratio, multiply 72 by
(__12 )
2
, n - 1 times. That is, you
n-1
.
The height of the ball on the 6th bounce is f 5 6-1 1 1 2 2 Multiply 72 by = .
(_ )
(_)
(
6
).
5
= 72(0.3125)
= 2.25 The height of the ball at the top of the 6th bounce is about
2.3
inches.
Reflect
7.
Is it possible for a sequence that describes the bounce height of a ball to have a common ratio greater than 1? No; if the common ratio were greater than 1, the bounce height would increase, which
© Houghton Mifflin Harcourt Publishing Company
1 (_12 ) = 72(_ 32 )
72
__1
could not happen in the real world.
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Your Turn
ELABORATE
8.
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Use examples to help students make
Physical Science A ball is dropped from a height of 8 meters. The table shows the height of each bounce. The heights form a geometric sequence. How high does the ball bounce on the 4th bounce? Round your answer to the nearest tenth of a meter. Bounce
The ball bounces about 2.5 meters on the 4th bounce.
9.
2
3
4
Term
1
2
4
8
Suppose all the terms of a geometric sequence are positive, and the common ratio r is between 0 and 1. Is the sequence increasing or decreasing? Explain. If r is between 0 and 1 and all the terms are positive, then each term is less than the
preceding term. So, the sequence is decreasing.
10. Essential Question Check-In If the common ratio of a geometric sequence is less than 0, what do you know about the signs of the terms of the sequence? Explain. The signs of the terms alternate. If r < 0 and the first term is negative, the second is © Houghton Mifflin Harcourt Publishing Company
1
Start with 1 and multiply each term by 2 to get the next term.
positive. Then the third must be negative. The signs of the terms continue to alternate. If the first term is positive, the results are similar.
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Elaborate
Term Number
Words
3.375
= 2.53125
Ways to Represent the Geometric Sequence 1, 2, 4, 8, …
n -1
3
f(4) = 6(0.75)
Complete the graphic organizer with students to discuss and summarize lesson concepts. In each box, write a way to represent the geometric sequence.
f(n) = 1(2)
4.5
6 4-1 f(4) = 6(0.75)
SUMMARIZE THE LESSON
Formula
6
2
4.5 _ = 0.75 = r
generalizations about geometric sequences of positive numbers for which the common ratio r is greater than 1 and geometric sequences of positive numbers for which the common ratio r is between 0 and 1, that is, 0 < r < 1.
Table
Height (m)
1
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Evaluate: Homework and Practice
EVALUATE • Online Homework • Hints and Help • Extra Practice
Find the common ratio r for each geometric sequence and use r to find the next three terms. 1.
3.
5, 15, 45, 135 … 15 =3=r 5 f(4) = 135, so f(5) = 135(3) = 405.
2.
_
-2, 6, -18, 54 … 6 = -3 = r -2 f(4) = 54, so f(5) = 54(-3) = -162.
_
f(6) = 405(3) = 1215
f(6) = 162(-3) = 486
f(7) = 1215(3) = 3645 The next three terms of the sequence are 405, 1215, and 3645.
f(7) = 486(-3) = -1458 The next three terms of the sequence are -162, 486, and -1458.
4, 20, 100, 500, …
4.
20 _ =5=r
4 f(4) = 500, so f(5) = 500(5) = 2500.
ASSIGNMENT GUIDE
8, 4, 2, 1, …
_4 = _1 = r
(_)
_(_) _ _(_) _
f(7) = 12,500(5) = 62,500 The next three terms of the sequence are 2500, 12,500, and 62,500.
_
The next three terms of the sequence are
_1, _1, and _1. 2 4
5.
72, -36, 18, -9, …
-36 1 =r _ = -_
6.
( _) ( _)
7.
8.
f(6) = 810(3) = 2430
Example 1 Extending Geometric Sequences
Exercises 1–12, 21–23
Example 2 Recognizing Growth Patterns of Geometric Sequences in Context
Exercises 13–20
_3 = 0.6 = r
f(6) = 0.648(0.6) = 0.3888
f(7) = 2430(3) = 7290 The next three terms of the sequence are 810, 2430, and 7290.
Exercise
5, 3, 1.8, 1.08, …
5 f(4) = 1.08, so f(5) = 1.08(0.6) = 0.648.
10 f(4) = 270, so f(5) = 270(3) = 810.
Exercises 1–10
-80 _ = -0.4 = r
f(7) = -2.048(-0.4) = 0.8192 The next three terms of the sequence are 5.12, -2.048, and 0.8192.
30 _ =3=r
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200, -80, 32, -12.8, …
f(6) = 5.12(-0.4) = -2.048
10, 30, 90, 270, …
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200 f(4) = -12.8, so f(5) = -12.8(-0.4) = 5.12.
4.5, -2.25, and 1.125.
Explore 1 Exploring Growth Patterns of Geometric Sequences
© Houghton Mifflin Harcourt Publishing Company
( _)
72 2 f(4) = -9, so f(5) = -9 - 1 = 4.5. 2 f(6) = 4.5 - 1 = -2.25 2 f(7) = -2.25 - 1 = 1.125 2 The next three terms of the sequence are
Practice
Explore 2 Comparing Growth Patterns of Arithmetic and Geometric Sequences
8 2 1 1 f(4) = 1, so f(5) = 1 = . 2 2 1 1 1 = f(6) = 4 2 2 1 1 1 ( ) f 7 = = 4 2 8
f(6) = 2500(5) = 12,500
Concepts and Skills
f(7) = 0.3888(0.6) = 0.23328 The next three terms of the sequence are 0.648, 0.3888, and 0.23328. Lesson 1
644
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1–10
1 Recall of Information
MP.6 Precision
11–14
2 Skills/Concepts
MP.5 Using Tools
15–20
1 Recall of Information
MP.5 Using Tools
21
2 Skills/Concepts
MP.2 Reasoning
22
3 Strategic Thinking
MP.5 Using Tools
23
3 Strategic Thinking
MP.3 Logic
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9.
AVOID COMMON ERRORS
18, 36, 72, 144
10. 243, 162, 108, 72, … 162 2 = =r 243 3 2 f(4) = 72, so f(5) = 72 = 48. 3 2 f(6) = 48 = 32 3 2 1 = 21 f(7) = 32 3 3 The next three terms of the sequence are 48, 32, and 21.3333.
_ _
36 _ =2=r
When finding a common ratio, students might divide in the wrong order. Make sure that students divide each term after the first term by the preceding term to find the common ratio.
18 f(4) = 144, so f(5) = 144(2) = 288
(_)
(_) (_)
f(6) = 288(2) = 576
f(7) = 576(2) = 1152 The next three terms of the sequence are 288, 576, and 1152.
_
Find the indicated term of each sequence by repeatedly multiplying the first term by the common ratio. Use a calculator. 11. 1, 8, 64, …; 5th term
12. 16, -3.2, 0.64, …; 7th term
8 =8=r 1 5-1 f(5) = 1(8)
16 7-1 f(7) = 16(-0.2)
_
= 1(8)
-3.2 _ = -0.2 = r
4
= 16(-0.2)
= 4096
= 0.001024
13. -50, 15, -4.5, …; 5th term
14. 3, -12, 48, …; 6th term
_
-12 _ = -4 = r
15 = -0.3 = r -50 5-1 f(5) = -50(-0.3) = -50(-0.3)
6
3 f(n) = f(1)r n - 1 f(6) = 3(-4)
4
= 3(-4)
= -0.405
6-1 5
= -3072
© Houghton Mifflin Harcourt Publishing Company
Solve. You may use a calculator and round your answer to the nearest tenth of a unit if necessary. 15. Physical Science A ball is dropped from a height of 900 centimeters. The table shows the height of each bounce. The heights form a geometric sequence. How high does the ball bounce on the 5th bounce?
Bounce
Height (cm)
1
800
2
560
3
392
560 _ = 0.7 = r
800 Find the 5th term of the sequence. f(5) = 800(0.7) = 800(0.7)
5-1 4
= 192.08 centimeters Module 14
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16. Leo’s bank balances at the end of months 1, 2, and 3 are $1500, $1530, and $1560.60, respectively. The balances form a geometric sequence. What will Leo’s balance be after 9 months? The first two terms of the sequence are 1500 and 1530. 1530 = 1.02 = r 1500 Find the 9th term of the sequence.
CRITICAL THINKING Have students explain why some sequences alternate signs and some do not. (When the common ratio, r, is negative, the terms in a geometric sequence alternate signs. When the common ratio is positive, the terms are all positive or all negative.)
_
f(9) = 1500(1.02) = 1500(1.02)
9-1 8
= $1757.49 17. Biology A biologist studying ants started on day 1 with a population of 1500 ants. On day 2, there were 3000 ants, and on day 3, there were 6000 ants. The increase in an ant population can be represented using a geometric sequence. What is the ant population on day 5?
3000 _ =2=r
1500 Find the 5th term of the sequence. f(5) = 1500 · 2 5-1 = 1500 · 2 4 = 24,000 ants
400 _ = 0.8 = r
Bounce
Height (cm)
1
500
2
400
3
320
500 Find the 8th term of the sequence. f(8) = 500(0.8) = 500(0.8)
8-1 7
= 104.8576 centimeters. 19. Finance The table shows the balance in an investment account after each month. The balances form a geometric sequence. What is the amount in the account after month 6? 2040 = 1.2 = r 1700 Find the 6th term of the sequence.
_
f(6) = 1700(1.2) = 1700(1.2)
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= $4230.14
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Month
Amount ($)
1
1700
2
2040
3
2448
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jolanta Dabrowska/Alamy
18. Physical Science A ball is dropped from a height of 625 centimeters. The table shows the height of each bounce. The heights form a geometric sequence. How high does the ball bounce on the 8th bounce?
6-1 5
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Understanding Geometric Sequences
646
20. Biology A turtle population grows in a manner that can be represented by a geometric sequence. Given the table of values, determine the turtle population after 6 years.
JOURNAL Have students explain the difference between the rules used to create the following sequences, and determine whether each sequence is arithmetic, geometric, or neither:
Year
Number of Turtles
1
5
2
15
3
45
15 _ =3=r
3, 6, 9, 12, 15, … arithmetic
5 Find the 6th term of the sequence.
3, 6, 10, 15, 21, … neither
f(6) = 5 · 3 6-1
3, 6, 12, 24, 48, … geometric
= 5 · 35 = 1215 turtles 21. Consider the geometric sequence –8, 16, –32, ... Select all that apply. a. The common ratio is 2.
16 ___ = -2, so the common ratio is -2. -8
b. The 5th term of the sequence is –128.
The common ratio is -2, and the 1st term is -8, f(n) = -8 (-2)
n-1
, and
f (5) = -8 (-2) = -128, and the 5th term of the sequence is -128. 4
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Olha K/ Shutterstock
c.
The 7th term is 4 times the 5th term.
The 7th term is -2 times the 6th term, which is -2 times the 5th term, so the 7th term is 4 times the 5th term. d. The 8th term is 1024.
The 8th term is -8 (-2) = 1024. 7
e. The 10th term is greater than the 9th term.
-8 (-2) is positive and -8 (-2) is negative, so the 10th term is greater than the 9th 9
8
term. So, B, C, D, and E all apply. H.O.T. Focus on Higher Order Thinking
22. Justify Reasoning Suppose you are given a sequence with r < 0. What do you know about the signs of the terms of the sequence? Explain. Because r < 0, if the first term is negative, the second is positive. Then the third must be negative. The signs of the terms continue to alternate. 23. Critique Reasoning Miguel writes the following: 8, x, 8, x, … He tells Alicia that he has written a geometric sequence and asks her to identify the value of x. Alicia says the value of x must be 8. Miguel says that Alicia is incorrect. Who is right? Explain. The value that Alicia gave is correct, but it is not the only correct value. x could also be -8. 8, -8, 8, -8, ... is also a geometric sequence, so Miguel is correct. Module 14
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Lesson Performance Task
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Guide students who need help in representing
Multi-Step Gifford earns money by shoveling snow for the winter. He offers two payment plans: either pay $400 for the entire winter or pay $5 for the first week, $10 for the second week, $20 for the third week, and so on. Explain why the two plans do or do not form a geometric sequence. Then determine how many weeks it will take for the plans to cost the same amount of money.
the geometric sequence 5, 10, 20, … by giving them an explicit function.
The two plans form a geometric sequence. In the first plan, the geometric sequence is a n = 400 ∗ 1 n. In the second plan,
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Have students explain how they could use
the geometric sequence is a n = 5∗ 2 n-1. To find out how long it will take the two plans to cost the same, set the two sides equal to each other and solve. 400∗ 1 n = 5 ∗ 2 n-1 400 = 5 · 2 n-1 80 = 2 n-1 2 5 = 32 2 6 = 64 2 7 = 128
reasoning to check their answers for the approximate number of weeks when the plans cost about the same amount of money. For example, some students may say they wrote out the geometric sequence until they could see that $400 would occur between weeks 7 and 8.
It will take between 6 weeks and 7 weeks for the plans to cost the same amount of money. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Trudy Wilkerson/Shutterstock
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Have students explain which of the following two options they would prefer. Have students explain their reasoning.
4/19/14 9:48 AM
Option 1: They receive 1¢ on the first day, 2¢ on the second day, 4¢ on the third day, 8¢ on the fourth day, and so on, doubling each day, for a total of 20 days. Option 2: They receive $5000 on the first day and $0 after that. Students will find that with Option 1, on the 20th day, without including the amounts from the previous 19 days, the amount received would be $5242.88, which is more than $5000. So, the total amount received in Option 1 would be much more than the total amount, $5000, received in Option 2.
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Understanding Geometric Sequences
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LESSON
14.2
Name
Constructing Geometric Sequences
Class
Date
14.2 Constructing Geometric Sequences Essential Question: How do you write a geometric sequence? Resource Locker
Common Core Math Standards The student is expected to: COMMON CORE
Explore 1
F-BF.A.1a
...Determine an explicit expression, a recursive process, or steps for calculation from a context. Also F-LE.A.2, F-BF.A.2
You learned previously that an explicit rule for a sequence defines the nth term as a function of n. A recursive rule defines the nth term of a sequence in terms of one or more previous terms.
Mathematical Practices COMMON CORE
You can use what you know to identify recursive and explicit rules for sequences, and identify whether the sequences are arithmetic, geometric, or neither.
MP.4 Modeling
The given rule is a(n)
A
Explain to a partner the difference between a recursive and an explicit rule for a geometric sequence.
value of
B
.
rule because you do not need to know the
.
n
The only unknown in the expression is
C
Essential Question: How do you write a geometric sequence?
The sequence is a(n) the
© Houghton Mifflin Harcourt Publishing Company
View the Engage section online. Discuss the photo and reasons why a rabbit population might change rapidly; for example, gestation is 31 days, litter size is 4 to 12, and rabbits mate as early as age 3 months. Then preview the Lesson Performance Task.
f(n − 1)
explicit
n-1
, which
represents the term number .
ENGAGE
PREVIEW: LESSON PERFORMANCE TASK
()
3 A rule for the sequence 6, 9, 13.5,... is ƒ(n) = 6 _ 2
Language Objective
Possible answer: You write down an initial term and then multiply each term by a common factor to get the next term.
Understanding Recursive and Explicit Rules for Sequences
product
geometric
sequence because each term is
of the previous term and __32 .
Reflect
1.
Discussion How can you differentiate between a geometric sequence and an arithmetic sequence? An arithmetic sequence involves addition (or subtraction) and a geometric sequence involves multiplication (or division).
2.
How can you tell by looking at a function rule for a sequence whether it is a recursive rule? If a given rule is recursive, the first term f(1) appears in the rule.
Module 14
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IN1_MNLESE389762_U6M14L2 649
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Lesson 2 649 Module 14
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Lesson 14.2
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10:06 AM
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Explain 1
Constructing Recursive and Explicit Rules for Given Geometric Sequences
EXPLORE
To write a recursive rule for a sequence, you need to know the first term, and a rule for successive terms. Example 1
Understanding Recursive and Explicit Rules for Sequences
Write a recursive rule and an explicit rule for each geometric sequence.
Makers of Japanese swords in the 1400s repeatedly folded and hammered the metal to form layers. The folding process increased the strength of the sword.
INTEGRATE TECHNOLOGY
The table shows how the number of layers depends on the number of folds.
Have students complete the Explore activity in either the book or online lesson.
Number of Folds Number of Layers
n
1
2
3
4
5
f(n)
2
4
8
16
32
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Ask students to explain the difference between a recursive rule and an explicit rule for a sequence, and tell how each one is written.
To write a recursive rule, find the common ratio by calculating the ratio of consecutive terms.
_4 = 2
2 The common ratio r is 2. The first term is 2, so f(1) = 2. ƒ(2) = ƒ(1) ∙ 2, ƒ(3) = ƒ(2) ∙ 2, ƒ(4) = ƒ(3) ∙ 2, . . . State the recursive rule by providing the first term and the rule for successive terms. ƒ(1) = 2 ƒ(n) = ƒ(n - 1) ∙ 2 for n ≥ 2 Write an explicit rule for the sequence by writing each term as the product of the first term and a power of the common ratio.
n 1 2 3 4 5
EXPLAIN 1
© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©kazenouta/Shutterstock
All terms after the first term are the product of the previous term and the common ratio:
f(n)
2 (2 ) = 2 0
2(2) = 4 1
2(2) = 8
Constructing Recursive and Explicit Rules for Given Geometric Sequences INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Ask students to explain what each of the values in the explicit rule in this Example represents. Ask them how the rule would change if various changes were made to the sequence.
2
2(2) = 16 3
2(2) = 32 4
Generalize the results from the table: ƒ(n) = 2 ∙ 2 n-1. Module 14
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Integrate Mathematical Practices
4/19/14 10:12 AM
This lesson provides an opportunity to address Mathematical Practice MP.4, which calls for students to use “modeling.” Students use a specific geometric sequence to help them obtain general rules, both recursive and explicit, for representing geometric sequences. They then use those general rules to represent other geometric sequences.
Constructing Geometric Sequences
650
B
QUESTIONING STRATEGIES
n f(n)
Which rule, the recursive rule or the explicit rule, would be more useful in determining the 20th term of a given sequence? Explain your reasoning. The explicit rule is more useful because you can easily evaluate f (20) to find the 20th term. To use the recursive rule, you would have to calculate each successive term until you get to the 20th term.
1
2
3
4
5
5
15
45
135
405
To write a recursive rule, find the common ratio by calculating the ratio of consecutive terms.
15 _= 3 5
3
The common ratio r is The first term is
5
. . So, the recursive rule is:
ƒ(n) = ƒ(n - 1) 3 for n ≥ 2 Write an explicit rule for the sequence by writing each term as the product of the first term and a power of the common ratio.
n
AVOID COMMON ERRORS Some students may confuse the recursive and explicit rules. Remind students that the recursive rule for a geometric sequence is stated by providing the first term, ƒ(1), and the rule for terms after the first term, so f(n – 1), should appear in this rule. An explicit rule for a geometric sequence is stated by writing each term as the product of the first term and a power of the common ratio.
f(n)
1
5(3) = 5
2
5(3) = 15
3
5(3) = 45
4
5(3) = 135
5
5(3) = 405
0
1
2
3
4
Generalize the results from the table: ƒ(n) = 5 · 3
n-1
.
© Houghton Mifflin Harcourt Publishing Company
Reflect
3.
Explain why the sequence 5, 10, 20, 40, 80, … appears to be a geometric sequence. The quotient of successive terms is always 2.
4.
Draw Conclusions How can you use properties of exponents to simplify the explicit rule ƒ(n) = 2 ∙ 2 n-1? n-1 Use the product of powers rule: 2 ∙ 2 = 2 1 ∙ 2 n-1 = 2 1+n-1 = 2 n. Therefore, the explicit rule can be simplified to f(n) = 2 n.
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Peer-to-Peer Activity Have students work in pairs. Have one student write an explicit rule for a geometric sequence and the other write a recursive rule for a different geometric sequence. Have students trade papers and write the first five terms of the sequence as well as the rule that is not already written, either explicit or recursive. Have students do this two or three times.
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Your Turn
EXPLAIN 2
Write a recursive rule and an explicit rule for each geometric sequence. 5.
n an
1
2
3
4
5
7
14
28
56
112
Deriving the General Forms of Geometric Sequence Rules
14 _ = 2 = r and the first term is 7.
7 Recursive rule: f(1) = 7 and f(n) = f(n - 1) ∙ 2 for n ≥ 2
QUESTIONING STRATEGIES
Explicit rule: f(n) = 7 ∙ 2 n-1 6.
If you know the second term and the common ratio of a geometric sequence, can you write an explicit rule for the sequence? If so, explain how. Yes; you can divide the second term by the common ratio to get the first term. Then, you can substitute the first term and the common ratio into the general explicit rule to get the explicit rule for the sequence.
Write a recursive rule and an explicit rule for the geometric sequence 128, 32, 8, 2, 0.5, ... . 32 = 0.25 = r and the first term is 128. 128 Recursive rule: f(1) = 128 and f(n) = f(n - 1) ∙ 0.25 for n ≥ 2
_
Explicit rule: f(n) = 128 ∙ 3(0.25) (n-1)
Explain 2 Example 2
Deriving the General Forms of Geometric Sequence Rules
Use each geometric sequence to help write a recursive rule and an explicit rule for any geometric sequence. For the general rules, the values of n are consecutive integers starting with 1.
6, 24, 96, 384, 1536, … Find the common ratio. Algebra
6, 24, 96, 384, 1536,…
ƒ(1), ƒ(2), ƒ(3), ƒ(4), ƒ(5),...
Common ratio = 4
Common ratio = r
© Houghton Mifflin Harcourt Publishing Company
Numbers
Write a recursive rule. Numbers
Algebra
ƒ(1) = 6 and
Given ƒ(1),
ƒ(n) = ƒ(n - 1) ∙ 4 for n ≥ 2
ƒ(n) = ƒ(n - 1) ∙ r for n ≥ 2
Write an explicit rule. Numbers
Algebra
ƒ(n) = 6 ∙ 4 n-1
ƒ(n) = ƒ(1) ∙ r n-1
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Kinesthetic Experience
4/19/14 10:12 AM
Students may benefit from acting out a situation that can be represented by a geometric sequence, such as folding a piece of paper in half repeatedly. The number of folds is the term number, n, and the number of layers is the term, ƒ(n). Have students write a recursive rule and an explicit rule for the sequence.
Constructing Geometric Sequences
652
B
VISUAL CUES
4, 12, 36, 108, 324,… Find the common ratio.
Have students color code the matching information found in the recursive rule and explicit rule of a geometric sequence as shown: Explicit Rule: f (n) = f (1) ∙ r
Numbers
Algebra
4, 12, 36, 108, 324,…
ƒ(1), ƒ(2), ƒ(3), ƒ(4), ƒ(5),...
Common ratio = 3
Common ratio =
r
Write a recursive rule. n−1
Recursive Rule: Given f (1), f (n) = f (n - 1) ∙ r .
Numbers
Algebra
ƒ(1) = 4 and
Given ƒ(1),
ƒ(n) = ƒ(n - 1) ∙ 3 for n ≥ 2
ƒ(n) = ƒ(n - 1) ∙
r for n ≥ 2
Write an explicit rule. Numbers ƒ(n) = 4 ∙ 3
Algebra n-1
( 1 )∙
ƒ(n) = ƒ
r
n-1
Reflect
7.
1 . Explain how the 4 th Discussion The first term of a geometric sequence is 81 and the common ratio is _ 3 term of the sequence can be determined. 1 The 4 th term of the sequence can be determined by raising to the 3 rd power and 3 multiplying this result by 81.
_
8.
What is the recursive rule for the sequence ƒ(n) = 5(4) ? The first term is 5 and the common ratio is 4. So a recursive rule is f 1 = 5, f(n) = f(n-1) ∙ 4 n-1
© Houghton Mifflin Harcourt Publishing Company
Your Turn
Use each geometric sequence to help write a recursive rule and an explicit rule for any geometric sequence. 9.
6, 12, 24, 48, 96,…
Find the common ratio. Numbers 6, 12, 24, 48, 96, … Common ratio = 2
Algebra
f(1), f(2), f(3), f(4), f(5), ...
Common ratio = r
Write a recursive rule. Numbers
Algebra
f(1) = 6 and
Given f(1),
f(n) = f(n - 1) ∙ 2 for n ≥ 2
f(n) = f(n-1) ∙ r for n ≥ 2
Write an explicit rule. Numbers
Algebra
f(n) = 6 ∙ 2 n-1
f(n) = f(1) ∙ r n-1
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Connect Vocabulary In this lesson, students learn about general rules, recursive rules, and explicit rules. Using the text, review with them and help them differentiate the meaning of each of these terms. Point out that the word recursive begins with the prefix re-, which means again. Tell them to think about the word repeat. Recursive refers to a procedure that can be repeated over and over. Speakers of Spanish will likely connect this with the prefix re- in Spanish, which has the same meaning. Words such as repetir (repeat) and reforzar (reinforce) are examples of Spanish words with the re- prefix.
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Explain 3
Constructing a Geometric Sequence Given Two Terms
EXPLAIN 3
The explicit and recursive rules for a geometric sequence can also be written in subscript notation. In subscript notation, the subscript indicates the position of the term in the sequence. a 1, a 2, and a 3 are the first, second, and third terms of a sequence respectively. In general, a n is the nth term of a sequence. Example 3
Constructing a Geometric Sequence Given Two Terms
Write an explicit rule for each sequence using subscript notation.
Photography The shutter speed settings on a camera form a geometric sequence where a n is the shutter speed in seconds and n is the setting number. The fifth setting on the 1 camera is __ second and the seventh setting on the camera 60 1 is __ second. 15
QUESTIONING STRATEGIES How is an explicit rule for a geometric sequence, written using function notation, related to an explicit rule for a geometric sequence, written using subscript notation? When the position numbers start with 1, the variable n is used in both notations to represent the nth term of the sequence. In both cases, the nth term is found by multiplying the first term by the common ratio raised to the power of (n - 1).
Identify the given terms in the sequence. 1 The fifth setting is __ second, so the 5th term of the 60 1 __ sequence is 60 .
1 a5 = _ 60 1 1 The seventh setting is __ second, so the 7th term of the sequence is __ . 15 15 1 a 7 = __ 15
Find the common ratio. Write the recursive rule for a 7.
a6 = a5 ∙ r
Write the recursive rule for a 6.
a7 = a5 ∙ r ∙ r
Substitute the expression for a 6 into the rule for a 7.
1 ∙ r2 1 =_ _ 15 60 4 = r2 2=r
© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©Itsra Sanprasert/Shutterstock
a7 = a6 ∙ r
1 for a and _ 1 for a . Substitute _ 7 5 15 60 Multiply both sides by 60. Definition of positive square root
Find the first term of the sequence. a n = a 1 ∙ r n-1
Write the explicit rule.
1 = a ∙ 2 5-1 _ 1 60 1 = a ∙ 16 _ 1 60 1 =a _ 1 960 Write the explicit rule.
1 for a , 2 for r, and 5 for n. Substitute _ n 60 Simplify.
a n = a 1 ∙ r n-1
Write the general rule.
1 ∙ (2)n-1 an = _ 960
1 for a and 2 for r. Substitute _ 1 960
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Divide both sides by 16.
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Constructing Geometric Sequences
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B
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Show students the relationship between the
Viral Video You tell a number of friends about an interesting video you saw online. Each of those friends tells the same number of friends about it. This pattern continues, and there are no repeats in the people told. The numbers of people who hear about this video through you form a geometric sequence. There are 256 people at the fourth round and 4096 people at the sixth round. Identify the given terms in the sequence.
terms of a geometric sequence using the following pattern:
The 4th term of the sequence is 256 . a 4 = 256
a1 = a1
The 6th term of the sequence is a6 =
a2 = a1 · r
4096
Find the common ratio.
a3 = a1 · r · r a4 = a1 · r · r · r Ask students to find other patterns that may be helpful when constructing a geometric sequence given two terms. For example:
a = a5 ∙ r
Write the recursive rule for a 6.
a = a4 ∙ r
Write the recursive rule for a 5.
a = a4 ∙
4096
a1 = a1
a2 = a1 · r
= 256 ∙ r 2
Substitute the expression for a 5 into the rule for a 6. Substitute
4096
for a 6 and 256 for a 4.
16 = r 2
Divide both sides by 256.
4 =r
Definition of positive square root.
a n = a 1 ∙ r n-1
4 -1
© Houghton Mifflin Harcourt Publishing Company
Write the explicit rule.
256 = a 1 ∙ 4
Substitute 256 for a n, 4 for r, and 4 for n.
256 = a 1 ∙ 64
Simplify.
4 = a1
Divide both sides by 64 .
Write the explicit rule. a n = a 1 ∙ r n-1 an = 4 ∙
(4)
n-1
Write the general rule. Substitute 4 for a 1 and 4 for r.
Reflect
10. Finding the common ratio in the shutter speed example involved finding a square root. Why was the negative square root not considered? The common ratio must be positive. The sequence represents shutter speeds, which must
be positive.
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r ∙r
Find the first term of the sequence.
a3 = a2 · r a4 = a3 · r
655
4096 .
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Lesson 2
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Your Turn
ELABORATE
Write an explicit rule for the sequence using subscript notation. 1 1 11. The third term of a geometric sequence is __ and the fifth term is ___ . All the terms of the sequence 27 243
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 In this lesson, students write a recursive rule
are positive.
Common ratio: a5 = a3 ∙ r2 1 1 = ∙ r2 243 27 1 r= 3
_ _ _
First term: 1 1 = a1 27 3
(_) 1 1 _ = a (_) 27 3 _
3-1
2
Write the explicit rule: n-1 1 1 an = 3 3
_(_)
1
_1 = a 3
and an explicit rule for a geometric sequence. Ensure that students understand what each value in the rules represents.
1
Elaborate
SUMMARIZE THE LESSON
12. What If Suppose you are given the terms a 3 and a 6 of a geometric sequence. How can you find the common ratio r? a6 Use the fact that a 6 = a 3 ∙ r ∙ r ∙ r = a 3 ∙ r 3 and find the cube root of a .
_
How can you write a recursive and an explicit rule for a geometric sequence? To write a recursive rule, assume f(1) is given and use the general rule f (n) = f (n−1) ∙r for each whole number n ≥ 2 , where r is the common ratio. To write an explicit rule, use the general rule f (n) = f (1) f (1)∙r n - 1, where f (1) is the first term in the sequence, r is the common ratio, and the domain of the function is the set of positive integers or a subset of consecutive positive integers beginning with 1.
3
Your Turn
13. If you know the second term and the common ratio of a geometric sequence, can you write an explicit rule for the sequence? If so, explain how. Yes; you can divide the second term by the common ratio to get the first term. Then, you
can substitute the first term and the common ratio into the general explicit rule to get the explicit rule for the sequence.
Use these values in the form f(n) = f(1) ∙ r n-1.
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© Houghton Mifflin Harcourt Publishing Company
14. Essential Question Check-In How can you write the explicit rule for a geometric sequence if you know the recursive rule for the sequence? Use the first term of the sequence and the common ratio as given in the recursive rule.
Lesson 2
4/19/14 10:12 AM
Constructing Geometric Sequences
656
Evaluate: Homework and Practice
EVALUATE
• Online Homework • Hints and Help • Extra Practice
For each geometric sequence, write a recursive rule by finding the common ratio by calculating the ratio of consecutive terms. Write an explicit rule for the sequence by writing each term as the product of the first term and a power of the common ratio. 1.
ASSIGNMENT GUIDE Concepts and Skills
n
1
2
3
4
5
an
2
6
18
54
162
_6 = 3 = r and a
1=2 2 n-1 Recursive rule: a 1 = 2 and a n = 3a n-1 for n ≥ 2; Explicit rule: a n = 2(3)
Practice
Explore Understanding Recursive and Explicit Rules for Sequences
2.
Example 1 Constructing Recursive and Explicit Rules for Given Geometric Sequences
Exercises 1–11
Example 2 Deriving the General Forms of Geometric Sequence Rules
Exercises 12–15, 27
Example 3 Constructing a Geometric Sequence Given Two Terms
Exercises 16–26, 28
n
1
2
3
4
5
an
10
3
0.9
0.27
0.081
3 _ = 0.3 = r and a
1 = 10 10 n-1 Recursive rule: a 1 = 10 and a n = 0.3a n-1 for n ≥ 2; Explicit rule: a n = 10(0.3)
3.
n
1
2
3
4
5
an
5
20
80
320
1280
20 _ = 4 = r and a
1=5 5 n-1 Recursive rule: a 1 = 5 and a n = 4a n-1 for n ≥ 2; Explicit rule: a n = 5(4)
© Houghton Mifflin Harcourt Publishing Company
4.
n
1
2
3
4
5
an
6
-3
1.5
-.75
.375
-3 1 = r and a _ = -_
1=6 6 2 Recursive rule: a 1 = 6 and a n = - 1 a n-1 for n ≥ 2; Explicit rule: a n = 6 - 1 2 2
5.
n an
1 9
2 6
_6 = _2 = r and a
Exercise
IN1_MNLESE389762_U6M14L2 657
Lesson 14.2
3
4
5
4
2 2_ 3
7 1_ 9
1=9 9 3 2 Recursive rule: a 1 = 9 and a n = 2 a n-1 for n ≥ 2; Explicit rule: a n = 9 3 3
(_ )
_
Module 14
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( _)
_
n-1
Lesson 2
657
Depth of Knowledge (D.O.K.)
n-1
COMMON CORE
Mathematical Practices
1–15
2 Recall of Information
MP.2 Reasoning
16–24
2 Skills/Concepts
MP.4 Modeling
25–26
3 Strategic Thinking
MP.4 Modeling
27
3 Strategic Thinking
MP.3 Logic
28
3 Strategic Thinking
MP.5 Using Tools
4/19/14 10:12 AM
6.
n
1
2
3
4
5
an
-12
6
-3
1.5
-0.75
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 After students have written the general
6 _ = -0.5= r and a
1 = -12 -12 n-1 Recursive rule: a 1 = -12 and a n = -0.5a n-1 for n ≥ 2; Explicit rule: a n = -12(-0.5)
7.
n
1
2
3
4
5
an
4
24
144
864
5184
algebraic formulas for recursive and explicit rules for geometric sequences, compare the process for substituting values for the variables in the formulas for geometric sequences with the process for arithmetic sequences. Point out the similarities such as the use of a common difference in arithmetic sequences and the use of a common ratio in geometric sequences. Also, point out the differences such as how the common difference is added in a recursive rule for an arithmetic sequence, while the common ratio is multiplied in a recursive rule for a geometric sequence.
24 _ = 6 = r and a
1=4 4 n-1 Recursive rule: a 1 = 4 and a n = 6a n-1 for n ≥ 2; Explicit rule: a n = 4(6)
8.
n
1
2
3
4
5
an
10
5
2.5
1.25
0.625
5 _ = 0.5 = r and a
1 = 10 10 n-1 Recursive rule: a 1 = 10 and a n = 0.5a n-1 for n ≥ 2; Explicit rule: a n = 10(0.5)
9.
n
1
2
3
4
5
an
3
21
147
1029
7203
21 _ = 7 = r and a
1=3 3 n-1 Recursive rule: a 1 = 3 and a n = 7a n-1 for n ≥ 2; Explicit rule: a n = 3(7)
a n = 3(7) 10.
n-1
1
2
3
4
5
an
8
72
648
5832
52,488
© Houghton Mifflin Harcourt Publishing Company
n
72 _ = 9 = r and a
1=8 8 n-1 Recursive rule: a 1 = 8 and a n = 9a n - 1 for n ≥ 2; Explicit rule: a n = 8(9)
11.
n
1
2
3
4
5
an
6
30
150
750
3750
30 _ = 5 = r and a
1=6 6 n-1 Recursive rule: a 1 = 6 and a n = 5a n - 1 for n ≥ 2; Explicit rule: a n = 6(5)
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Lesson 2
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Constructing Geometric Sequences
658
Use the geometric sequence to help write a recursive rule and an explicit rule for any geometric sequence. For the general rules, the values of n are consecutive integers starting with 1.
AVOID COMMON ERRORS Some students may confuse the recursive and explicit rules for geometric sequences. Remind students that the recursive rule for a geometric sequence is stated by providing the first term, ƒ(1), and a rule for terms after the first term in terms of the previous term. Such a rule has the general form ƒ(n) = ƒ(n−1)∙ r , where r is the common ratio. An explicit rule for a geometric sequence is stated by indicating the domain of the sequence and giving a rule for the nth term, with the general form ƒ(n)= ƒ(1)∙ r n - 1 , where r is the common ratio. Show students examples.
12. 5, 15, 45, 135, 405,…
Numbers Common ratio:
5, 15, 45, 135, 405,… Common ratio = 3
Recursive rule:
f(1) = 5 and
f(n) = f(n - 1) ∙ 3 for n ≥ 2 Explicit rule:
f(n) = 5 ∙ 3 n-1
10, 40, 160, 640, 2560,… Common ratio = 4
Recursive rule: Write an explicit rule:
f(n) = f(n - 1) ∙ r for n ≥ 2
f(n) = f(1) ∙ r n-1
Algebra f(1), f(2), f(3), f(4), f(5),...
Common ratio = r
f(1) = 10 and
Given f(1),
f(n) = 10 ∙ 4 n-1
f(n) = f(1) ∙ r n-1
f(n) = f(n - 1) ∙ r for n ≥ 2
14. 5, 10, 20, 40, 80,…
© Houghton Mifflin Harcourt Publishing Company
Numbers Common ratio:
5, 10, 40, 80, 160,…
Recursive rule:
f(1) = 5 and
Explicit rule:
Algebra f(1), f(2), f(3), f(4), f(5),...
Common ratio = 2
Common ratio = 2
f(n) = f(n - 1) ∙ 2 for n ≥ 2
f(n) = f(n - 1) ∙ r for n ≥ 2
f(n) = 5 ∙ 2
n-1
Given f(1),
f(n) = f(1) ∙ r n-1
15. 18, 90, 450, 2250, 11,250,…
Numbers Common ratio: Recursive rule: Explicit rule:
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Lesson 14.2
Given f(1),
Numbers Common ratio:
COMMUNICATING MATH
659
f(1), f(2), f(3), f(4), f(5),...
Common ratio = r
13. 10, 40, 160, 640, 2,560,…
f(n) = f(n - 1) ∙ 4 for n ≥ 2
Tell students that they have learned about two kinds of sequences in this course: arithmetic and geometric. Arithmetic sequences have common differences and geometric sequences have common ratios. Common ratios are found by dividing each term after the first term by the term before it. Have students state in their own words how to find any term in a geometric sequence.
Algebra
18, 90, 450, 2250, 11,250,…
Algebra f(1), f(2), f(3), f(4), f(5),...
Common ratio = 5
Common ratio = r
f(n) = f(n - 1) ∙ 5 for n ≥ 2
f(n) = f(n - 1) ∙ 5 for n ≥ 2
f(1) = 18 and f(n) = 18 ∙ 5
n-1
659
Given f(1),
f(n) = f(1) ∙ r n-1
Lesson 2
4/19/14 10:12 AM
Write an explicit rule for each geometric sequence using subscript notation. Use a calculator and round your answer to the nearest tenth if necessary. 16. The fifth term of the sequence is 5. The sixth term is 2.5.
Common ratio: a 6 = a 5 ∙ r
76.8 = 120 ∙ r 2
76.8 _ =r
2
120 0.64 = r 2
First term: 2.5 = a 1(0.5)
5
2.5 = a 1(0.03125) 80 = a 1
r = 0.8
First term: 76.8 = a 1(0.8)
Equation: a n = 80(0.5)
4
76.8 = a 1(0.4096)
n-1
187.5 = a 1
Equation: a n = 187.5(0.64) 18. The fourth term of the sequence is 216. The sixth term is 96.
96 = 216 ∙ r 2 4 = r2 9 2 r= 3
_
2 First term: 96 = a (_) 1
Common ratio: a 6 = a 4 ∙ r 2
8 = 32 ∙ r 2
8 _ =r 32 1 r=_
5
2
3
32 (_ 243 )
96 = a 1
729 = a 1
Equation: a n = 729
2
1 First term: 8 = a (_) 1
n-1
8 = a1
256 = a 1
5
2
1 (_ 32 )
Equation: a n = 256
(_12 )
n-1
20. Video Games The number of points that a player must accumulate to reach the next level of a video game forms a geometric sequence where a n is the number of points needed to complete level n. You need 20,000 points to complete level 3 and 8,000,000 points to compete level 5.
Common ratio: a5 = a3 ∙ r
2
First term: 8,000,000 = a 1(20)
Write the equation: 4
8,000,000 = 20,000 ∙ r 2
8,000,000 = a 1(160,000)
400 = r
50 = a 1
2
a n = 50(20)
n-1
© Houghton Mifflin Harcourt Publishing Company
(_23 )
n-1
19. Sports The numbers of teams in a singleelimination tennis tournament represents a geometric sequence where a n is the number of teams competing and n is the round. There are 32 teams remaining in round 4 and 8 teams in round 6.
Common ratio: a 6 = a 4 ∙ r 2
_
In the explicit rule of a function, can you substitute any term in the sequence for a 1? No; the correct terms would not be generated if a term other than a 1 were substituted.
Common ratio: a 5 = a 3 ∙ r 2
2.5 = 5 ∙ r r = 0.5
QUESTIONING STRATEGIES
17. The third term of the sequence is 120. The fifth term is 76.8.
r = 20 Module 14
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21. Conservation A state began an effort to increase the deer population. In year 2 of the effort, the deer population in a state forest was 1200. In year 4, the population was 1728.
JOURNAL Have students write the first five terms of a geometric sequence for which r is greater than 1. Then have them write the first five terms of a geometric sequence for which r is less than 1. Have the students explain how to write the explicit and recursive rules for each sequence.
First term:
Common ratio: a4 = a2 ∙ r
(_65 ) 216 1728 = a (_) 125 5
1728 = a 1
2
1728 = 1200 ∙ r 2 1728 = r2 1200 6 r= 5
_ _
Equation: a n = 1000
1
1000 = a 1
(_65 )
n-1
22. Biology The growth of a local raccoon population approximates a geometric sequence where a n is the number of raccoons in a given year and n is the year. After 6 years there are 45 raccoons and after 8 years there are 71 raccoons.
First term:
Common ratio: a8 = a7 ∙ r
45 = a 1(1.26)
a8 = a6 ∙ r2
14 = a 1
© Houghton Mifflin Harcourt Publishing Company ©Alan D Carey/Photodisc/ Getty Images
71 = 45 ∙ r 2
n-1
1.58 = r 2
r = 1.26
23. Chemistry A chemist measures the temperature in degrees Fahrenheit of a chemical compound every hour. The temperatures approximate a geometric sequence where a n is the temperature at a given hour, and n is the hour. At hour 4, the temperature is 70 °F and after 5 hours the temperature is 80 °F.
First term:
Common ratio:
80 = a 1(1.07)
a6 = a4 ∙ r2
Write the equation: a n = 57(1.07)
5
80 = a 1(1.40255)
80 = 70 ∙ r 2 80 = r2 70 1.14 = r 2
_
57 ≈ a 1
r = 1.07
24. Yusuf was asked to write a recursive rule for a sequence. Which of the following is an appropriate answer? Select all that apply. a. f(n) = 11(5)
n-1
d. f(n) = 12 + 19 ∙ f(n - 1)
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Lesson 14.2
a n = 14(1.26)
45 = 3.18a 1
a7 = a6 ∙ r
661
Write the equation: 6-1
b. f(n) = 11f(n), f(1) = 555
()
2 e. f(n) = -4 _ 3
661
n-1
c. f(n) = f(n) + 15, f(1) = 36
So only b and c are recursive.
Lesson 2
4/19/14 10:12 AM
H.O.T. Focus on Higher Order Thinking
QUESTIONING STRATEGIES
25. Multi-Step An economist predicts that the cost of food will increase by 4% per year for the next several years.
Is there a common difference in the sequence of population values? If so, what is it? no
a. Write an explicit rule for the sequence that gives the cost f(n) in dollars of a box of cereal in year n that costs $3.20 in year 1. Justify your answer. f(n) = 3.2 ∙ 1.04 n - 1 ; the cost in year 1, f(n), is $3.20. An increase of 4% means the cost is multiplied by 1.04, that is, f(n) = f(n) ∙ 1.04 b. What is the fourth term of the sequence? What does it represent in this situation? Justify your answer. $3.60; it represents the cost of the box of cereal in year 4. f(4) = 3.2 ∙ 1.04 3 = 3.6 26. Analyze Relationships Suppose you know the 8th term of a geometric sequence and the common ratio r. How can you find the 3rd term of the sequence without writing a rule for the sequence? Explain. Divide the 8th term by the common ratio 5 times. If you were given the 3rd term, you would multiply it by the common ratio 5 times to get the 8th term, so perform the inverse operation. 27. Explain the Error Given that the second term of a sequence is 64 and the fourth n-1 1 for the sequence. Explain his error. term is 16, Francis wrote the explicit rule a n = 128 ∙ _ 4 Francis used the square of the common ratio. For the given sequence, a 2 = 64, and a 4 = 16,
Is there a common ratio in the sequence of population values? If so, what is it? yes; 1.5
INTEGRATED MATHEMATICAL PRACTICES Focus on Technology MP.5 Students can use a computer spreadsheet to extend the geometric sequence for this situation by following these instructions. Enter 800 in cell A1. Then click on cell A2 and input the formula, =A1*1.5, into the formula window and touch Enter. The value 1200 will appear in cell A2. Place your cursor on the bottom right hand corner of cell A2 so that a plus sign appears. Click the plus sign and drag the cursor down 10 or more cells below A2. The successive terms of the sequence will appear.
()
1 1 so 16 = 64 · r 2. Then r 2 = _ and r = _ . 4 2
28. Communicate Mathematical Ideas Suppose you are given two terms of a geometric sequence like the ones in Example 2, except that both terms are negative. Explain how writing the explicit rule for the sequence would differ from the examples in this lesson.
Lesson Performance Task The table shows how a population of rabbits has changed over time. Write an explicit rule for the geometric sequence described in the table. In what year will there be more than 5000 rabbits?
Time (years), n
Population, a n
1
800
2
1200
3
1800
4
2700
_
1200 = 1.5 = r 800
a n = 800 * 1.5
In(6.25) = (n - 1)In(1.5) 1.83 = .41(n - 1)
n-1
1.83 = .41n - .41
5000 = 800 * 1.5 n - 1
2.24 = .41n
6.25 = 1.5
5.46 = n
n-1
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There will be more than 5000 rabbits after about 5.5 years.
© Houghton Mifflin Harcourt Publishing Company (t) ©Plush Studios/Blend Images/Corbis; (b) ©amanaimages/Corbis
When you find the common ratio r, you have to consider both the positive and negative values. The term between the given terms might be negative or it might be positive. If it is positive, then the common ratio r is negative. If it is negative, then r is positive. You would have to consider both possibilities. If r > 0, then the first term must be negative. If r < 0, then the first term must be positive.
Lesson 2
EXTENSION ACTIVITY IN1_MNLESE389762_U6M14L2 662
Have students investigate the Fibonacci sequence, a sequence that is neither arithmetic or geometric. The first two terms of the sequence are 1 and 1. Each term after that is generated by adding the previous two terms. The sequence is 1, 1, 2, 3, 5, 8, 13, …. Have students write a recursive rule for this sequence and extend the sequence for at least 4 more terms.
A recursive rule is a 1 = 1, a 2 = 1, a n = a n - 2 + a n - 1 for n ≥ 3. The next 4 terms are 21, 34, 55, 89.
4/19/14 10:12 AM
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Constructing Geometric Sequences
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LESSON
14.3
Name
Constructing Exponential Functions
Class
Date
14.3 Constructing Exponential Functions Essential Question: What are discrete exponential functions and how do you represent them? Resource Locker
Common Core Math Standards The student is expected to: COMMON CORE
Explore
F-LE.A.2
Construct… geometric... sequences, given a graph, a description of a relationship, or two input-output pairs… Also F-IF.A.2, F-IF.C.7e
Recall that a discrete function has a graph consisting of isolated points.
Mathematical Practices COMMON CORE
MP.6 Precision
The table represents the cost of tickets to an annual event as a function of the number t of tickets purchased. Complete the table by adding 10 to each successive cost. Plot each ordered pair from the table.
Language Objective Explain to a partner what the graph of a discrete exponential function looks like.
PREVIEW: LESSON PERFORMANCE TASK
Tickets t
Cost ($)
(t, f(t))
1
10
(1, 10)
2
20
(2, 20)
3
30
4
40
5
50
( ( (
3, 30 4, 40 5, 50
) ) )
y
60 50 Cost ($)
Possible answer: Discrete exponential functions are exponential functions whose domains are limited, such as the set of integers, making the graph appear as isolated points instead of a continuous curve.
© Houghton Mifflin Harcourt Publishing Company
ENGAGE Essential Question: What are discrete exponential functions and how do you represent them?
Understanding Discrete Exponential Functions
40 30 20 10 x
View the Engage section online. Discuss the photo and why the population of invasive species might grow exponentially. Then preview the Lesson Performance Task.
0
1
2
3
4
5
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Module 14
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Lesson 3
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Date Class
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HARDCOVER PAGES 663676
ing Disc
IN1_MNLESE389762_U6M14L3 663
Explore
Understand
. isolated points
of consisting er t of has a graph of the numb d function a function each ordere a discrete l event as cost. Plot an annua successive tickets to 10 to each the cost of by adding represents lete the table The table ased. Comp tickets purch (t, f(t)) the table. pair from Cost ($) (1, 10) Tickets t 10 1 (2, 20) 20 2 3, 30 30 3 4, 40
Watch for the hardcover student edition page numbers for this lesson.
Recall that
(
40
4 5
y g Compan
(
)
(
)
5, 50
50
)
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60
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Harcour t
Cost ($)
Publishin
50 40 30
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20 x
10 0
1
3
2
4
5
Tickets Lesson 3 663
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Lesson 14.3
4/19/14
10:23 AM
4/19/14 10:25 AM
B
The number of people attending an event doubles each year. The table represents the total attendance at each annual event as a function of the event number n. Complete the table by multiplying each successive attendance by 2. Plot each ordered pair from the table.
Event Number n
Attendance
(n, g(n))
1
20
(1, 20)
2
40
(2, 40)
3
80
4
160
5
320
360
( ( (
3, 80 4, 160 5, 320
EXPLORE Understanding Discrete Exponential Functions
) ) )
INTEGRATE TECHNOLOGY Have students complete the Explore activity in either the book or online lesson.
QUESTIONING STRATEGIES
y
How are the tables for the functions different? They are different because successive output values of the first function have a common difference, while successive output values of the second function have a common ratio.
320
Attendance
280 240 200 160 120 80 © Houghton Mifflin Harcourt Publishing Company
40 x 0
1
2
3
4
5
Event Number
C
Complete the table.
Function
Linear?
Discrete?
f(t)
Yes
Yes
g(n)
No
Yes
Reflect
1.
Communicate Mathematical Ideas What are the limitations on the domains of these functions? Why? The domains include only whole numbers. You cannot buy a fraction of a ticket to an event, nor stage a fraction of an event.
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Lesson 3
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U6M14L3 664
Learning Progressions
4/19/14 10:25 AM
In this lesson, students learn how to identify and represent exponential functions. Some key understandings for students are as follows: • An exponential function can be represented by an equation of the form f (x)= abx, where a, b, and x are real numbers, a ≠ 0, b > 0, and b ≠ 1. • For an exponential function ƒ(x), if x is in the domain of the function, then the ratio of ƒ(x+1) to ƒ(x) is a constant ratio. In the next module, students will learn more about exponential functions, including exponential growth and decay functions.
Constructing Exponential Functions
664
Representing Discrete Exponential Functions
Explain 1
EXPLAIN 1
An exponential function is a function whose successive output values are related by a constant ratio. An exponential function can be represented by an equation of the form ƒ(x) = ab x, where a, b, and c are real numbers, a ≠ 0, b > 0, and b ≠ 1. The constant ratio is the base b.
Representing Discrete Exponential Functions
When evaluating exponential functions, you will need to use the properties of exponents, including zero and negative exponents. Recall that, for any nonzero number c: c 0 = 1, c ≠ 0
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Point out that the constant b in a function of
1 , c ≠ 0. c -n = _ cn Example 1
the form f (x) = abx, where a and b are real numbers, a ≠ 0, b > 0, and b ≠ 1, corresponds to the common ratio, r, in a geometric sequence. Explain to students that if the domain of the function is the set of positive integers, the points of the graph of a discrete exponential function represent a geometric sequence. The x-values are the term numbers of the sequence and the y-values are the corresponding terms.
2 = ∙ __ 2 = 2. reciprocal of the denominator: 1· __ 1 1 2
© Houghton Mifflin Harcourt Publishing Company
__1
( )
1 x with a domain of ⎧⎨-2, -1, 0, 1, 2, 3, 4⎫⎬ ƒ(x) = 3 ⋅ _ ⎭ ⎩ 2 -1 1 = 3⋅ _ 1 = 3⋅2= 6 ƒ(-1) = 3 ⋅ _ 2 1 _ 2
QUESTIONING STRATEGIES How do you evaluate a complex fraction, such 1 ? Multiply the numerator by the as __
Complete the table for each function using the given domain. Then graph the function using the ordered pairs from the table.
( )
()
(x, f(x))
x
f(x)
-1
6
(-1, 6)
0
3
(0, 3)
1
1 1_ 2
2
3 _ 4
3
3 _ 8
4
3 _ 16
(1, 1_12 ) (2, _34 ) (3, _38 ) 3 (4, _ 16 )
y 6 5 4 3 2 1 x -1
0
1
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Lesson 3
COLLABORATIVE LEARNING IN1_MNLESE389762_U6M14L3 665
Peer-to-Peer Activity
Have students work in pairs. Instruct one student in each pair to construct a table of values and graph for the function defined by h(x) = 3x with the domain ⎧ ⎫ ⎨-2, -1, 0, 2⎬. Instruct the other student to do the same for the function defined ⎩ ⎭ by g(x) = 3x with the same domain. Have the students compare and contrast the tables of values and the graphs.
665
Lesson 14.3
4/19/14 10:25 AM
B
( )
4 x; domain = ⎧⎨-2, -1, 0, 1, 2⎫⎬ ƒ(x) = 3 _ ⎩ ⎭ 3
()
4 ƒ(-3) = 3 _ 3
-2
( ) 3
x
-2
27 11 9 3 1 = 3 ⋅ _ = _ = 1_ = 3⋅ _ =3⋅ _ -2 -2 16 16 16 4 4 _ -2
f(x)
-2
11 1__ 16
-1
1 2_ 4
0
3
1
4
2
1 5_ 3
3
1 7_ 9
( (
(x, f(x))
) _ ) ( ) ( ) ( _)
__ -2, 1 16 11
-1,
214
0,
3
1,
4
2,
513
(3, 7_19 )
y 8 6 4 © Houghton Mifflin Harcourt Publishing Company
2 x -4
-2
0
2
4
Reflect
2.
What If What would happen to the function ƒ(x) = ab x if a were 0? What if b were 1? If a = 0 then ƒ(x) = 0 for all x, and if b = 1, then ƒ(x) = a for all x. Neither of these constant functions are exponential functions.
3.
Discussion Why is a geometric sequence a discrete exponential function? A sequence is a function. When the input values increase by 1, the successive output values are related by a constant ratio, the common ratio. The general recursive rule for a geometric sequence is the form of an exponential function.
Module 14
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Lesson 3
DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U6M14L3 666
Cognitive Strategies
4/19/14 10:25 AM
Encourage students to isolate the necessary information from any problem scenario they are given. It may help to circle or underline values or to write the important information on a separate sheet of paper.
Constructing Exponential Functions
666
Your Turn
EXPLAIN 2
Make a table for the function using the given domain. Then graph the function using the ordered pairs from the table.
Constructing Exponential Functions from Verbal Descriptions
4.
()
x
3 ; domain = ⎧⎨-3, -2, -1, 0, 1, 2⎫⎬ (x) = 4 _ ⎩ ⎭ 2 y
x
8
MODELING
-3
6
Have students fold a piece of paper in half repeatedly. Have them make a table representing the number of layers of paper after 0 folds, 1 fold, 2 folds, and so on.
-2
4
-1
2 x -4
-2
0
2
4
QUESTIONING STRATEGIES How do you find the value of b in order to write an exponential function in the form ƒ(x) = abx ? Choose a number x in the domain of the function. Then divide f(x + 1) by f(x) to get b. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Digital Vision/Getty Images
Explain 2
f(x)
32 5 = 1_ (_ 27 ) 27 16 7 = 1_ (_ 9) 9 8 2 _ _ (3) = 23
(x, f(x))
5 (-3, 1_ 27 ) (-2, 1_79 ) (-1, 2_23 )
0
4
(0, 4)
1
6
(1, 6)
2
9
(2, 9)
Constructing Exponential Functions from Verbal Descriptions
You can write an equation for an exponential function by finding or calculating the values of a and b. The value of a is ƒ(x + 1) the value of the function when x = 0. The value of b is the common ratio of successive function values, b = _______ . ƒ(x) For discrete functions with integer or whole number domains, these will be successive values of the function. Example 2
Write an equation for the function.
When a piece of paper is folded in half, the total thickness doubles. Suppose an unfolded piece of paper is 1 millimeter thick. The total thickness t(n) of the paper is an exponential function of the number of folds n. The value of a is the original thickness of the paper before any folds are made, or 1 millimeter. Because the thickness doubles with each fold, the value of b (the constant ratio) is 2. The equation for the function is t(n) = 0.1(2) . n
Module 14
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Lesson 3
LANGUAGE SUPPORT IN1_MNLESE389762_U6M14L3 667
Connect Vocabulary In this lesson there are several key words that begin with the prefix in-. Learning the meaning of prefixes helps all students of math and science, since many key terms in both disciplines originated in Latin or have Latin prefixes. In this lesson, the words increase and input both start with the prefix in-, which means in, into, on, near, or towards. The prefix in- can also mean not, as in the word inverse. Another key prefix in this lesson is non-, which means not. The term nonzero, for example, means not zero.
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Lesson 14.3
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A savings account with an initial balance of $1000 earns 1% interest per month. That means that the account balance grows by a factor of 1.01 each month if no deposits or withdrawals are made. The account balance in dollars B(t) is an exponential function of the time t in months after the initial deposit.
EXPLAIN 3 Constructing Exponential Functions from Input-Output Pairs
Let B represent the balance in dollars as a function of time t in months.
1000 . The value of a is the original balance, The value of b is the factor by which the balance changes every month,
) ( The equation for the function is B(t) = 1000 1.01 .
1.01 .
t
AVOID COMMON ERRORS When calculating the value of b, students may divide in the wrong order. Remind them that they should always divide an output value by the preceding output value. For example, if they are using (4, 27) and (5, 9), they should divide the function value of the second pair by the function value of the first pair:
Reflect
5.
Why is the exponential function in the paper-folding example discrete? The paper can only be folded a whole number of times.
Your Turn
6.
A piece of paper that is 2 millimeters thick is folded. Write an equation for the thickness t of the paper in millimeters as a function of the number n of folds.
Initial thickness (n = 0)
a = 0.2
Change per fold: double
b=2 t(n) = 0.2 ⋅ 2 n
Equation:
Explain 3
9 = __ 1. b = ___ 27 3
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 After students have found an equation for an
Constructing Exponential Functions from Input-Output Pairs
You can use given two successive values of a discrete exponential function to write an equation for the function. Example 3
(3, 12) and (4, 24)
24 = 2. Find b by dividing the function value of the second pair by the function value of the first: b = _ 12 Evaluate the function for x = 3 and solve for a.
Write the general form.
ƒ(x) = ab x
Substitute the value for b.
ƒ(x) = a ⋅ 2 x
Substitute a pair of input-output values.
12 = a ⋅ 2 3
Simplify.
12 = a ⋅ 8
Solve for a. Use a and b to write an equation for the function.
Module 14
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© Houghton Mifflin Harcourt Publishing Company
exponential function, have them check both pairs in the equation to verify that the equation is correct.
Write an equation for the function that includes the points.
3 a= _ 2 3 ⋅ 2x ƒ(x) = _ 2
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Constructing Exponential Functions
668
( )
9 (1, 3) and 2, _ 4
B
QUESTIONING STRATEGIES
9 ÷3= Find b by dividing the function value of the second pair by the first: b = _ 4 Write the general form. ƒ ( x) = a b x
If an exponential function includes the points (4, 64) and (5, 32), what is the common ratio? Explain how you found it. The common ratio is 0.5. 64 = __ 1 = 0.5 . I found it by dividing 32 by 64: ___ 32 2
Substitute the value for b. Substitute a pair of input-output values.
ELABORATE CRITICAL THINKING
3= a⋅
Solve for a.
a= ƒ(x) =
_3 4
4
()
3 4 _ 4
x
(-2, _52 ) and (-1, 2)
2 =5 b=2÷_ 5 a 2 = a ⋅ 5 -1; 2 = _; a = 10 5 f(x) = 10 ⋅ 5 x
Elaborate
© Houghton Mifflin Harcourt Publishing Company
8.
Explain why the following statement is true: For 0 < b < 1 and a > 0, the function ƒ(x) = ab decreases as x increases. When you multiply a number a by a fraction b between 0 and 1, the product is less than x
the original number a. When you multiply by this fraction b repeatedly, the results decrease further. 9.
Explain why the following statement is true: For b > 1 and a > 0, the function ƒ(x) = ab increases as x increases. When you multiply a number a by a number b greater than 1, the product is greater x
than the original number a. When you multiply by this number b repeatedly, the results increase further. 10. Essential Question Check-In What property do all pairs of adjacent points of a discrete exponential function share? The ratio of function values is the same for any two pairs.
Module 14
IN1_MNLESE389762_U6M14L3 669
Lesson 14.3
1
( )
Your Turn
SUMMARIZE THE LESSON
669
x
(_34)
Write an equation for the function that includes the points. 7.
What are discrete exponential functions and how can you represent them? A discrete exponential function is a function of the form f (x) = abx , with a ≠ 0, b > 1, and b ≠ 1, such that the graph of the function is a set of isolated points. For example, the domain of the function could be the set of whole numbers. Such a function can be represented by a set of input and output values in a table or by a graph of ordered pairs.
.
4
3 3 = a⋅ _ 4
Simplify.
Use a and b to write an equation for the function.
Have students give an example of an increasing exponential function and of a decreasing exponential function.
ƒ ( x) = a ⋅
_3
669
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Evaluate: Homework and Practice
EVALUATE • Online Homework • Hints and Help • Extra Practice
Complete the table for each function using the given domain. Then graph the function using the ordered pairs from the table. 1.
1 ⋅ 4 x; domain = ⎧⎨-2, -1, 0, 1, 2⎫⎬. ƒ(x) = _ ⎩ ⎭ 2
x
y
8
-2
6
2.
-2
2
0
_1
1
2
2
8
0
2
4
()
(x, f(x) ) (-2, __1 ) 32
ASSIGNMENT GUIDE
(-1, _18) (0, _12)
_1
4
8 2
(1, 2) (2, 8)
x
1 ; domain = ⎧⎨0, 1, 2, 3, 4, 5⎫⎬. ƒ(x) = 9 _ ⎩ ⎭ 3
x
f(x)
8
0
9
6
1
3
4
2
1
2
3
_1
4
_1
5
1 __
y
x 0
2
4
6
8
()
(2, 1)
(3, _13) (4, _19) 1 (5, __ 27 )
3 9
27
2 ; domain = ⎧⎨-1, 0, 1, 2, 3, 4⎫⎬. ƒ(x) = 6 _ ⎩ ⎭ 3
x
f(x)
8
-1
9
6
0
6
4
1
4
2
2
2 2_ 3
3
7 1_ 9
4
5 1__ 27
x 0
2
4
6
8
Module 14
Exercise
IN1_MNLESE389762_U6M14L3 670
(0, 6)
COMMON CORE
(1, 4)
(2, 2_23) (3, 1_79) 5 (4, 1__ 27 )
Mathematical Practices
1 Recall of Information
MP.6 Precision
5–8
2 Skills/Concepts
MP.4 Modeling
9–21
2 Skills/Concepts
MP.2 Reasoning
1 Recall of Information
MP.2 Reasoning
3 Strategic Thinking
MP.3 Logic
23–24
Practice
Explore Understanding Discrete Exponential Functions
Exercise 19
Example 1 Representing Discrete Exponential Functions
Exercises 1–4, 22–23
Example 2 Constructing Exponential Functions from Verbal Descriptions
Exercises 5–8, 22–21
Example 3 Constructing Exponential Functions from Input-Output Pairs
Exercises 9–18, 24
Lesson 3
1–4
22
(x, f(x)) (-1, 9)
670
Depth of Knowledge (D.O.K.)
(0, 9) (1, 3)
x
y
(x, f(x))
Concepts and Skills
© Houghton Mifflin Harcourt Publishing Company
3.
1 __ 32
-1
x -4
f(x)
4/19/14 10:24 AM
Constructing Exponential Functions
670
4.
QUESTIONING STRATEGIES
()
x
4 ; domain = ⎧⎨-3, -2, -1, 0, 1⎫⎬ ƒ(x) = 6 _ ⎩ ⎭ 3
Why is the restriction b ≠ 1 in the definition of an exponential function necessary? What would happen if b = 1? 1 raised to any power is 1, so the function would always have the same value. It would not increase or decrease.
8
y
6 4 2 x -8
-6
-4
-2
0
(x, f(x)) 17 ) (-3, 2__
x
f(x)
-3
17 2__ 32
-2
3 3_
-1
1 4_ 2
0
6
(0, 6)
1
8
(1, 8)
8
32
(-2, 3_38) (-1, 4_12)
Write an equation for each function. 5.
Business A recent trend in advertising is viral marketing. The goal is to convince viewers to share an amusing advertisement by e-mail or social networking. Imagine that the video is sent to 100 people on day 1. Each person agrees to send the video to 5 people the next day, and to request that each of those people send it to 5 people. The number of viewers v(n) is an exponential function of the number n of days since the video was first shown. a = 100 b=5 v(n) = 100 ⋅ 5 n
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©David J. Green - technology/Alamy
6.
A pharmaceutical company is testing a new antibiotic. The number of bacteria present in a sample when the antibiotic is applied is 100,000. Each hour, the number of bacteria present decreases by half. The number of bacteria remaining r(n) is an exponential function of the number n of hours since the antibiotic was applied.
a = 100,000 1 b=_ 2
7.
a=5
b = 0.99, (power remaining after 1% lost)
p(n) = 5(0.99)
IN1_MNLESE389762_U6M14L3 671
Lesson 14.3
n
Optics A laser beam with an output of 5 milliwatts is directed into a series of mirrors. The laser beam loses 1% of its power every time it reflects off of a mirror. The power p(n) is a function of the number of reflections.
Module 14
671
()
1 r(n) = 100,000 _ 2
n
671
Lesson 3
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8.
The NCAA basketball tournament begins with 64 teams, and after each round, half the teams are eliminated. The number of remaining teams t(n) is an exponential function of the number n of rounds already played. a = 64 1 b=_ 2
AVOID COMMON ERRORS Students may forget to find the value of both a and b when writing a function rule for an exponential function of the form ƒ(x) = ab x. Remind students that they need to determine the common ratio and the initial value of an exponential function in order to write the function rule.
()
1 t(n) = 64 _ 2
n
Write an equation for the function that includes the points. 9.
(2, 100) and (3, 1000)
10. (-2, 4) and (-1, 8) 8 b= 4 =2
_
1000 b = ____ 100
= 10
8 = a ⋅ 2 -1 a 8= 2 a = 16
100 = a ⋅ 10 2
_
100 = 100a a=1
f(x) = 16 ⋅ 2 x
f(x) = 10 x 11.
(1, _54 ) and (2, _32 )
12.
(_23) (_45)
(-3, _161 ) and (-2, _83 ) (_38) 1 (__ 16 )
b = ___
b =___
2 _ =_ ⋅5 3 4
3 __ =_ ⋅ 16 8 1
_4 = a ⋅ (_5)1 5 6 _4 = a ⋅ _5 5 6 4 _ a=_ ⋅6 5 5 24 = __ 25 x 24 _ f(x) = __ ⋅ (5)
1 __ = a ⋅ 6 -3 16 1 1 __ = a ⋅ ___ 16 216 1 a = __ ⋅ 216
5 =_ 6
16
= 13.5
© Houghton Mifflin Harcourt Publishing Company
25
=6
f(x) = 13.5 ⋅ 6 x
6
Use two points to write an equation for the function. 13.
x
f(x)
1
2
2
2 _ 7
3
2 _ 49
4
2 _ 343
Module 14
IN1_MNLESE389762_U6M14L3 672
(_27)
b = ___ 2 1 =_ 7
()
1 2=a⋅ _ 7
1
1 2=a⋅_ 7
a = 14 1 f(x) = 14 ⋅ _ 7
()
672
x
Lesson 3
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Constructing Exponential Functions
672
Use points (-2, 53) and (-1, 530)
14.
x
f(x)
-4
0.53
-3
5.3
-2
53
-1
530
530 b = ___ 53
= 10
530 = a ⋅ (10)
-1
1 530 = a ⋅ __ 10
a = 5300
f(x) = 5300 ⋅ (10)
x
15. 8
Use points (0, 3) and (1, 6)
y
6 b=_ 3
(1, 6)
6
=2
4 2
a=3
(0, 3)
f(x) = 3 ⋅ 2 x
x -6
-4
0
-2
2
16. 8
y
Use points (1, 8) and (2, 4)
(1, 8)
4 b=_ 8 1 =_ 2
6 4
© Houghton Mifflin Harcourt Publishing Company
2 x 0
2
4
()
1 8=a _ 2
(2, 4)
6
8
a 8=_ 2
1
a = 16 f(x) = 16 ⋅ 2 x
17. The height h(n) of a bouncing ball is an exponential function of the number n of bounces. One ball is dropped and on the first bounce reaches a height of 6 feet. On the second bounce it reaches a height of 4 feet. 4 b=_ 6 2 =_ 3
()
2 6=a _ 3
1
2 6=a⋅_ 3
3 a=6⋅_ 2
=9
()
2 h(n) = 9 ⋅ _ 3 Module 14
IN1_MNLESE389762_U6M14L3 673
673
Lesson 14.3
n
673
Lesson 3
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18. A child starts a playground swing from standing and doesn’t use her legs to keep swinging. On the first swing she swings forward by 18 degrees, and on the second swing she only comes 13.5 degrees forward. The measure in degrees of the angle m(n) is an exponential function of the number of swings.
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Point out to students that, unlike linear
13.5 b = ___ 18 3 =_ 4
functions, the letter b in an exponential function of the form ƒ(x) = ab x does not represent the y-intercept.
()
3 18 = a _ 4
1
3 18 = a ⋅ _ 4
4 a = 18 ⋅ _ 3
= 24 3 m(n) = 24 _ 4
()
n
19. Make a Prediction A town’s population has been declining in recent years. The table shows the population since 1980. Is this data consistent with an exponential function? Explain. If so, predict the population for 2010 assuming the trend holds.
Year
Population
1980
5000
1990
4000
2000
3200
2010
2560 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Steve Hix/ Somos Images/Corbis
To check for exponential behavior, find the ratios between successive values of the function. 4000 4 ____ 4 ____ =_ , 3200 = _ 5000
5 4000
5
4 The ratio between successive function values is _ , so this is consistent with 5
an exponential function. 4 3200 ⋅ _ = 2560 5
20. A piece of paper has a thickness of 1.5 millimeters. Write an equation to describe the thickness t(n) of the paper when it is repeatedly folded in thirds.
b=3 a = 0.15 t(n) = 0.15 ⋅ 3 n
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Lesson 3
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Constructing Exponential Functions
674
21. Probability The probability of getting heads on a single coin flip is __12 . The probability of getting nothing but heads on a series of coin flips decreases by __12 for each additional coin flip. Write an exponential function for the probability of getting all heads in a series of n coin flips.
JOURNAL Have students describe a specific discrete exponential function in at least two different ways. They can describe it with words, a function rule, a table, and/or a graph.
1 b=_ 2 1 1 =a _ 2 2
_
()
1
_1 = a ⋅ _1
2 2 a=1 n p ( n) = 1 2
(_)
22. Multipart Classification Determine whether each of the functions is exponential or not. a. ƒ(x) = x 2
b. ƒ(x) = 3 ⋅ 2 x 1 c. ƒ(x) = 3 ⋅ _ x 2 d. ƒ(x) = 1.001 x
e. ƒ(x) = 2 ⋅ x 3 1 f. ƒ(x) = _ ⋅ 5 x 10
Exponential
Not exponential
Exponential
Not exponential
Exponential
Not exponential
Exponential
Not exponential
Exponential
Not exponential
Exponential
Not exponential
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©mj007/ Shutterstock
H.O.T. Focus on Higher Order Thinking
23. Explain the Error Biff observes that in every math test he has taken this year, he has scored 2 points higher than the previous test. His score on the first test was 56. He models his test scores with the exponential function s(n) = 28 · 2 n where s(n) is the score on his nth test. Is this a reasonable model based? Explain.
No; Biff’s test scores do not increase at a constant rate. Instead they go up by 2 points each test. His scores are not consistent with an exponential function. 24. Find the Error Kaylee needed to write the equation of an exponential function from points on the graph of the function. To determine the value of b, Kaylee chose the ordered pairs (1, 6) and (3, 54) and divided 54 by 6. She determined that the value of b was 9. What error did Kaylee make?
She should use ordered pairs with x-values that differ by 1. The value Kaylee found is not the constant ratio of successive values.
Module 14
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675
Lesson 14.3
675
Lesson 3
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Lesson Performance Task
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have students explain why they would choose
In ecology, an invasive species is a plant or animal species newly introduced to an ecosystem, often by human activity. Because invasive species often lack predators in their new habitat, their populations typically experience exponential growth. A small initial population grows to a large population that drastically alters an ecosystem. Feral rabbits that populate Australia and zebra mussels in the Great Lakes are two examples of problematic invasive species that grew exponentially from a small initial population.
3 or its decimal equivalent, 1.5, to use the fraction __ 2 when solving Part C of the Lesson Performance Task.
An ecologist monitoring a local stream has been collecting samples of an unfamiliar fish species over the past four years and has summarized the data in the table.
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 After students have completed the graph of
Here are the results so far:
Year
Average Population Per Mile
2009
32
2010
48
2011
72
2012
108
the populations versus time, discuss how the value of b, when a is positive, causes the graph to increase over time. Then discuss what happens to the graph t when students let b = __23 in f (t) = 32 __23 .
2013 2014
()
a. Look at the data in the table and confirm that the growth pattern is exponential.
48 _ = 1.5 32
72 _ = 1.5 48
108 _ = 1.5 72
Since there is a common ratio, r = 1.5, the growth pattern is exponential. b. Write the equation that represents the average population per square meter as a function of years since 2009.
()
t 3 The function is f(t) = 32 _ . 2
© Houghton Mifflin Harcourt Publishing Company
c. Predict the average populations expected for 2013 and 2014.
2013: 162 individuals per mile d. Graph the population versus time since 2009 and include the predicted values.
Average population per mile
2014: 243 individuals per mile 250 P 200 150 100 50 t 0
1
2
3
4
5
Time (years)
Module 14
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Lesson 3
EXTENSION ACTIVITY IN1_MNLESE389762_U6M14L3 676
Have students research the feral rabbit population in Australia or the zebra mussel population in the Great Lakes. Have students find out how the population was introduced into the new habitat and describe efforts to control the population.
4/19/14 10:24 AM
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Constructing Exponential Functions
676
LESSON
14.4
Name
Graphing Exponential Functions
Class
Date
14.4 Graphing Exponential Functions Essential Question: How do you graph an exponential function of the form f(x) = ab x?
Common Core Math Standards The student is expected to: COMMON CORE
Exponential functions follow the general shape y = ab x.
Graph exponential… functions, showing intercepts and end behavior… Also F-IF.C.8b
Mathematical Practices COMMON CORE
Exploring Graphs of Exponential Functions
Explore
F-IF.C.7e
Resource Locker
Graph the exponential functions on a graphing calculator, and match the graph to the correct function rule. 1.
MP.4 Modeling
2.
Language Objective
3. 4.
Explain the domain, range, and end behavior of the graphs of exponential functions of the form ƒ(x) = ab x with a < 0 and 0 < b < 1.
y = 3(2)
x
y = 0.5(2) y = 3(0.5)
a x
c
x
b
y = -3(2)
x
a.
b.
c.
d.
d
Essential Question: How do you graph an exponential function of the form f(x) = abx? Use the value of a to find the y-intercept. Choose several values of x other than 0 to plot a few more ordered pairs and connect them with a smooth curve. Use the values of a and b to determine the end behavior of the function.
© Houghton Mifflin Harcourt Publishing Company
ENGAGE
In all the functions 1–4 above, the base b > 0. Use the graphs to make a conjecture: State the domain and range of y = ab x if a > 0.
In all of the functions 1–4 above, the domain values are all real numbers, or -∞ < x < ∞. If a > 0 and b > 0, then the range values are all positive, or y > 0.
In all the functions 1–4 above, the base b > 0. Use the graphs to make a conjecture: State the domain and range of y = ab x if a < 0.
In all of the functions 1–4 above, the domain values are all real numbers, or -∞ < x < ∞. If a < 0 and b > 0, then the range values are all negative, or y < 0.
PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and the fact that the weight of a pumpkin might grow exponentially. Then preview the Lesson Performance Task.
What is the y-intercept of ƒ(x) = 0.5(2) ? x
0.5 Module 14
Lesson 4
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IN1_MNLESE389762_U6M14L4 677
Exploring
Explore
follow functions Exponential
the
HARDCOVER PAGES 677690
Graphs
y= general shape
on a functions exponential Graph the on rule. correct functi a. x a (2) 1. y = 3 x c (2) 2. y = 0.5 x b (0.5) 3. y = 3 x d (2) 4. y = -3
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Watch for the hardcover student edition page numbers for this lesson.
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d. c.
Harcour t
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x > 0. b > 0. = ab if a , the base x < ∞. range of y 1–4 above or -∞ < domain and functions numbers, State the are all real a conjecture: in values s to make e, the doma Use the graph or y > 0. 1–4 abov all positive, functions values are In all of the the range b > 0, then If a > 0 and 0. 0. x > < b = ab if a , the base range of y 1–4 above functions domain and numbers, In all the State the are all real a conjecture: or y < 0. in values s to make all negative, e, the doma Use the graph 1–4 abov values are functions the range In all of the b > 0, then a < 0 and x < ∞. If or -∞ < x (2) ? (x) = 0.5 ƒ of y-intercept Lesson 4 What is the
In all the
© Houghto
n Mifflin
0.5 Module 14
ESE3897
4L4 677 62_U6M1
IN1_MNL
677
Lesson 14.4
677 4/19/14
11:18 AM
4/19/14 11:20 AM
Note the similarities between the y-intercept and a. What is their relationship?
EXPLORE
The value of a in y = ab x is the y-intercept.
Exploring Graphs of Exponential Functions
Reflect
1.
Discussion What is the domain for any exponential function y = ab x? all real numbers, or -∞ < x < ∞.
2.
Discussion Describe the values of b for all functions y = ab x The base b is positive, or b > 0, in functions of the form y = ab x
Explain 1
INTEGRATE TECHNOLOGY Have students complete the Explore activity in either the book or online lesson.
Graphing Increasing Positive Exponential Functions
The symbol ∞ represents infinity. We can describe the end behavior of a function by describing what happens to the function values as x approaches positive infinity (x → ∞) and as x approaches negative infinity (x → -∞). Example 1
QUESTIONING STRATEGIES
Graph each exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph. Use inequalities to discuss the behavior of the graph.
What is the relationship between the value of a and the y-intercept of the graph of ƒ(x) = ab x ? The y-intercept is equal to a.
ƒ(x) = 2 x Choose several values of x and generate ordered pairs.
x -1
f(x) = 2 x
EXPLAIN 1
0.5
0
1
1
2
2
4
Graphing Increasing Positive Exponential Functions
Graph the ordered pairs and connect them with a smooth curve.
y
f(x) = 2x (2, 4)
4
b=2
2 (-1, 0.5)
y-intercept: (0, 1) End Behavior: As x-values approach positive infinity (x → ∞), y-values approach positive infinity (y → ∞). As x-values approach negative infinity (x → -∞), y-values approach zero (y → 0).
-4
-2
0
(1, 2) (0, 1) 2
x 4
-2 -4
Using symbols only, we say: As x → ∞, y → ∞, and as x → -∞, y → 0.
Module 14
678
© Houghton Mifflin Harcourt Publishing Company
a=1
AVOID COMMON ERRORS When generating ordered pairs for exponential functions of the form ƒ(x) = ab x, students may multiply a by b and then raise that product to x. Remind students to use the correct order of operations.
Lesson 4
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U6M14L4 678
Integrate Mathematical Practices
4/19/14 11:20 AM
This lesson provides an opportunity to address Mathematical Practice MP.4, which calls for students to use “modeling.” Students use tables and graphs to represent exponential functions. They can use the graphs to determine the end behavior of the functions, and make generalizations about the effect of parameters a and b on the end behavior of an exponential function of the form f(x) = ab x.
Graphing Exponential Functions 678
ƒ(x) = 3(4)
B
QUESTIONING STRATEGIES
x
Choose several values of x and generate ordered pairs.
If an exponential function is of the form ƒ(x) = ab x, both a and b are positive real numbers, and the domain is the set of all real numbers, what is the range of the function? The range is set of all positive real numbers.
x -1
f(x) = 3(4)
x
0.75 3 12 48
0 1 2
Graph the ordered pairs and connect them with a smooth curve.
y 45
b= 4
30
f(x) = 3(4)x
15 (-1, 0.75)
x
y-intercept:
(0,3)
End Behavior: As x → ∞, y → ∞ and as x → -∞, y → 0 .
-4
-2
(1, 12) (0, 3) 0 2 4
-15
Reflect
3.
(2, 48)
a= 3
If a > 0 and b > 1, what is the end behavior of the graph? If a > 0 and b > 1, as x approaches infinity y approaches infinity, and as x approaches
negative infinity y approaches 0. 4.
Describe the y-intercept of the exponential function ƒ(x) = ab x in terms of a and b. A graph of the exponential f(x) = ab x will have a y-intercept at (0, a).
Your Turn
5.
ƒ(x) = 2(2)
© Houghton Mifflin Harcourt Publishing Company
x -1 0 1 2
x
f(x) = 2(2)
x
9
1 2 4 8
6 (-1, 1) -4
a=2
b=2
y-intercept: (0, 2)
-2
3 0
y (2, 8) f(x) = 2(2)x (1, 4) (0, 2) 2
x 4
-3 -6
End Behavior: As x →∞, y →∞ and as x → -∞, y → 0.
Module 14
679
Lesson 4
COLLABORATIVE LEARNING IN1_MNLESE389762_U6M14L4 679
Peer-to-Peer Activity In pairs, have students develop and record their own rules for determining whether a graph represents one of the four main types of exponential functions: increasing positive, decreasing negative, decreasing positive, and increasing negative. For example, the rules might be a > 0 and b > 1, a < 0 and 0 < b < 1, a > 0 and 0 < b < 1, and a < 0 and b > 1. Then, give each pair one of each type of function and its graph. Have the students use their rules to determine which type of exponential function each graph represents. As a class, discuss the rules that worked.
679
Lesson 14.4
4/19/14 11:20 AM
Graphing Decreasing Negative Exponential Functions
Explain 2
EXPLAIN 2
You can use end behavior to discuss the behavior of a graph. Example 2
Graphing Decreasing Negative Exponential Functions
Graph each exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph. Use end behavior to discuss the behavior of the graph.
ƒ(x) = -2(3)
x
QUESTIONING STRATEGIES
Choose several values of x and generate ordered pairs.
x -1
f(x) = -2(3) -0.7 -2
0
-6
1
-18
2
Graph the ordered pairs and connect them with a smooth curve.
y
a = -2
6
b=3
x -4
y-intercept: (0, -2)
ƒ(x) = -3(4)
-1 1 2
x
Graph the ordered pairs and connect them with a smooth curve. a = -3 b= 4 y-intercept:
(2, -18)
40
y
20
( 0 , -3 )
End Behavior: As x → ∞, y → -∞ and as x → -∞, y → 0 .
f(x) = -3(4)x (0, -3) -4 -2 0 2 4 (1, -12) -20 -40 -60
Module 14
680
x
© Houghton Mifflin Harcourt Publishing Company
0
f(x) = -3(4) -0.75 -3 -12 -48
Students may forget that by the order of operations x rules, ƒ(x) = -2 x means that ƒ(x) = -1(2 ), and therefore the negative sign is not raised to the power. Tell students that the base of an exponential function cannot be negative.
f(x) = -2(3)x
-18
Choose several values of x and generate ordered pairs.
AVOID COMMON ERRORS
2 4 (1, -6)
-12
x
x
-2
(0, -2) -6
End Behavior: As x → ∞, y → -∞ and as x → -∞, y → 0.
How can the value of a tell you in which quadrants the graph of an exponential function in the form ƒ(x) = ab x is located? If a is greater than 0, the graph is located in quadrants I and II. If a is less than 0, the graph is located in quadrants III and IV.
x
(2, -48)
Lesson 4
DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U6M14L4 680
Cognitive Strategies
4/19/14 11:20 AM
Guide students to see that for exponential functions of the form f (x) = ab x , b > 1 represents growth (the graph is increasing), and b < 1 represents decline (the graph is decreasing). Remind students that if b = 1, the function is linear and not exponential; the function represents no change: no growth or decline.
Graphing Exponential Functions
680
Reflect
EXPLAIN 3
6.
Graphing Decreasing Positive Exponential Functions
If a < 0 and b > 1, what is the end behavior of the graph? If a < 0 and b > 1, as x approaches infinity y approaches negative infinity, and as
x approaches negative infinity y approaches 0. Your Turn
7.
QUESTIONING STRATEGIES
ƒ(x) = -3(3)
x
f(x) = -3(3)
x
If 0 < b < 1 and a is a positive constant, how can you alter b to make the graph of y = ab x decrease more gradually? Increase the value of b.
-1 0 1 2
x
20
-1 -3 -9 -27
y
10 -4
a = -3
(0, -3) 2 4 (1, -9) -10
-2
-20
b=3
f(x) = -3(3)x (2, -27)
-30
y-intercept: (0, -3)
x
0
End Behavior: As x → ∞, y → -∞ and as x → -∞, y → 0.
Explain 3
Graphing Decreasing Positive Exponential Functions
Graph each exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph. Use inequalities to discuss the behavior of the graph.
A
ƒ(x) = (0.5)
x
© Houghton Mifflin Harcourt Publishing Company
Choose several values of x and generate ordered pairs.
x -1
f(x) = (0.5) 2
0
1
1
0.5
2
0.25
Graph the ordered pairs and connect them with a smooth curve. a=1
y f(x) = 0.5x
4
(-1, 2) 2 (1, 0.5) (2, 0.25) (0, 1)
b = 0.5
-4
y-intercept: (0, 1) End Behavior: As x → ∞, y → 0 and as x → -∞, y → ∞.
Module 14
x
681
-2
0
2
x
4
-2 -4
Lesson 4
LANGUAGE SUPPORT IN1_MNLESE389762_U6M14L4 681
Connect Vocabulary Make sure that students understand what is meant by the end behavior of a function. The end behavior of a function describes what happens to the function values as x gets larger and larger (for example, as x becomes 100, 1000, and 1,000,000) and what happens to the function values as the values of x get smaller and smaller (for example, as x becomes –100, –1000, and –1,000,000). So, end behavior describes what happens to f(x) when x is farther and farther to the right and what happens to f(x) when x is farther and farther to the left.
681
Lesson 14.4
4/19/14 11:20 AM
B
ƒ(x) = 2(0.4)
x
COGNITIVE STRATEGIES
Choose several values of x and generate ordered pairs.
x -1
f(x) = 2(0.4) 5 2
0
()
1 y = 3 __ 2
0.8 0.32
1 2
Graph the ordered pairs and connect them with a smooth curve. a= 2 b = .4 y-intercept:
f(x) = 2(0.4)x
-4
-2
x
0
2
x 4
-2 -4
Reflect
8.
and y = 3(2) .
4 2 (0, 2) (1, 0.8)
End Behavior: As x → ∞, y → 0 and as x → -∞, y → ∞ .
x
y
6 (-1, 5)
(0,2)
( )
x
1 is the Show students that the graph of y = a __ b reflection of y = ab x over the y-axis by graphing
x
If a > 0 and 0 < b < 1, what is the end behavior of the graph? If a > 0 and 0 < b < 1, as x approaches infinity y approaches zero, and as x approaches
negative infinity y approaches infinity. Your Turn
9.
Graph the exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph. Use inequalities to discuss the behavior of the graph. ƒ(x) = 3(0.5) x
x
x
(-1, 6) 6 f(x) = 3(0.5)x 4 (0, 3) 2
6 3 1.5 0.75
-4
a=3
b = 0.5
-2
0
y
(1, 1.5) (2, 0.75) 2
x
4
-2 -4
y-intercept: (0, 3)
End Behavior: As x →∞, y → 0 and as x → -∞, y → ∞.
Module 14
IN1_MNLESE389762_U6M14L4 682
682
© Houghton Mifflin Harcourt Publishing Company
-1 0 1 2
f(x) = 3(0.5)
Lesson 4
4/19/14 11:20 AM
Graphing Exponential Functions
682
Graphing Increasing Negative Exponential Functions
Explain 4
EXPLAIN 4
Graph each exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph. Use inequalities to discuss the behavior of the graph.
Graphing Increasing Negative Exponential Functions
Example 4
ƒ(x) = -0.5 x Choose several values of x and generate ordered pairs.
QUESTIONING STRATEGIES
f(x) = -0.5 x
x
-2
-1
Why does it make sense for the graph of f (x) = ab x to stretch when a > 1, and for the graph to shrink when 0 < a < 1, compared to the graph of f (x) = b x ? Multiplying by a factor a greater than 1 increases the outputs, compared with the function f (x) = b x, and multiplying by a factor a between 0 and 1 decreases the outputs, compared with the function f (x) = b x.
-1
0
-0.5
1
-0.25
2
Graph the ordered pairs and connect them with a smooth curve.
2
a=1
1
b = 0.5
y-intercept: (0, 1)
-2
End Behavior: As x → ∞, y → 0 and as x → -∞, y → -∞.
ƒ(x) = -3(0.4)
x
-1
f(x) = -3(0.4)
x
-3
2
© Houghton Mifflin Harcourt Publishing Company
683
Lesson 14.4
x
-1.2 -0.48
1
Graph the ordered pairs and connect them with a smooth curve.
6
a = -3
y
3
(
0 , -3
)
x -2
End Behavior: As x → ∞, y → 0 and as x → -∞, y → -∞ .
IN1_MNLESE389762_U6M14L4 683
1
-3
0
Module 14
0
-1
-7.5
-1
y-intercept:
(2, -0.25) x
(1, -0.5) (0, -1) f(x) = -0.5x (-1, -2) -2
Choose several values of x and generate ordered pairs.
b = 0.4
y
683
-1
0
1
2 (1, -1.2) -3 (0, -3) f(x) = -3(0.4)x -6 (-1, -7.5) -9
Lesson 4
4/19/14 11:20 AM
Reflect
INTEGRATE TECHNOLOGY
10. If a < 0 and 0 < b < 1, what is the end behavior of the graph? If a < 0 and 0 < b < 1, as x approaches infinity y approaches zero, and as x approaches
Using graphing calculators can allow students to compare the graphs of exponential functions quickly, but they must enter the functions
negative infinity y approaches negative infinity. Your Turn
()
11. Graph the exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph. Use inequalities to discuss the behavior of the graph. x ƒ(x) = -2(0.5)
x -1 0 1 2
f(x) = -2(0.5)
x
4
-4 -2 -1 -0.5
y
2
ELABORATE
x -2
-1
0
2 (1, -1) -2 (0, -2) f(x) = -2(0.5)x
a = -2
(-1, -4)
b = 0.5
y-intercept: (0, -2)
1
QUESTIONING STRATEGIES
-4
Which of the following functions has the greatest y-intercept and which has the least 1 (2.5)x, y = 2 (2.5)x, y-intercept: y = 2.5 x, y = __ 2 x x or y = 4(2.5) ? Explain. y = 4 (2.5) has the 1 (2.5)x has the least greatest y-intercept (4), and y = __ 2 1 . y-intercept __ 2
-6
End Behavior: As x → ∞, y → 0 and as x → -∞, y → -∞.
Elaborate
()
12. Why is ƒ(x) = 3(-0.5) not an exponential function? x f(x) = 3(-0.5) is not an exponential function because its values alternate between x
negative and positive, creating a line that jumps between increasingly smaller positive
that takes on increasingly smaller positive or negative values as x approaches infinity.
13. Essential Question Check-In When an exponential function of the form ƒ(x) = ab x is graphed, what does a represent? a represents the y-intercept of the function.
SUMMARIZE THE LESSON
© Houghton Mifflin Harcourt Publishing Company
and negative values as x approaches infinity. An exponential function has a smooth curve
Copy and complete the graphic organizer with your students. In each box, give an example of an appropriate exponential function and sketch its graph.
Exponential Functions: y = ab x a > 0, b > 1 y = 3(2)x
IN1_MNLESE389762_U6M14L4 684
684
a < 0, b > 1 y = -1(3)x
y
12
Module 14
()
x
2 is correctly. Show students that y = -3 ∗ __ 3 x 2 by graphing not the same as y = -3 ∗ __ 3 both functions on a calculator.
y
-3
x
Lesson 4
-3
4/19/14 11:20 AM
0
2
12
0
a < 0, 0 < b < 1 1 x 3
y = -2
y
x
y -3
x -3
3
x
3
-12
3
a > 0, 0 < b < 1 x y=2 1
0
0
3
-12
Graphing Exponential Functions
684
Evaluate: Homework and Practice
EVALUATE
State a, b, and the y-intercept then graph the function on a graphing calculator. ƒ(x) = 2(3)
1.
x
2.
ƒ(x) = -6(2)
x
Graph:
Graph:
a=2
a = -6
y-intercept: (0, 2)
y-intercept: (0, -6)
• Online Homework • Hints and Help • Extra Practice
ASSIGNMENT GUIDE Concepts and Skills
Practice
Explore Exploring Graphs of Exponential Functions
Exercises 1–2, 19
Example 1 Graphing Increasing Positive Exponential Functions
Exercises 5, 9, 13, 15, 18, 20–22
Example 2 Graphing Decreasing Negative Exponential Functions
Exercises 8, 12, 16, 23
Example 3 Graphing Decreasing Positive Exponential Functions
Exercises 4, 7, 10, 17, 25
b=2
b=3 ƒ(x) = -5(0.5)
3.
x
Graph:
a = -5
a=3
On a graphing calculator, have students graph
()
x
()
x
x x 1 , and y = -3 __ 1 . y = 3(2) , y = -3(2) , y = 3 __ 2 2 Have them keep track of the changes by drawing sketches of each graph on the same coordinate grid.
© Houghton Mifflin Harcourt Publishing Company
INTEGRATE TECHNOLOGY
ƒ(x) = 6(3)
5.
x
y-intercept: (0, 3) 6.
ƒ(x) = -4(0.2)
x
Graph:
Graph:
a=6
a = -4
y-intercept: (0, 6)
y-intercept: (0, -4)
b=3
Module 14
Exercise
IN1_MNLESE389762_U6M14L4 685
b = 0.2
Lesson 4
685
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1–8
1 Recall of Information
MP.2 Reasoning
9–18
2 Skills/Concepts
MP.2 Reasoning
2 Recall of Information
MP.5 Using Tools
20–21
1 Skills/Concepts
MP.4 Modeling
22–25
3 Strategic Thinking
MP.3 Logic
19
Lesson 14.4
x
b = 0.8
y-intercept: (0, -5)
Exercises 3, 6, 11, 14, 24
ƒ(x) = 3(0.8)
Graph:
b=5
Example 4 Graphing Increasing Negative Exponential Functions
685
4.
4/19/14 11:20 AM
7.
ƒ(x) = 7(0.9)
x
8.
ƒ(x) = -3(2)
QUESTIONING STRATEGIES
x
Graph:
Graph:
a=7
a = -3
y-intercept: (0, 7)
y-intercept: (0,-3)
b = 0.9
How do you know the exponent in ab x applies to b and not ab? By the order of operations, bases are raised to exponents before multiplying.
b=2
State a, b, and the y-intercept then graph the function and describe the end behavior of the graphs. 9.
ƒ(x) = 3(3)
10. ƒ(x) = 5(0.6)
x
Ordered pairs:
x
Ordered pairs: f(x) = 3(3)
x -1 0 1 2
x
1 3 9 27
-1 0 1 2
Graph the ordered pairs and connect them with a smooth curve.
30
y f(x) = 3(3)x
-1
9 6 (0, 5) 3
(1, 9) x
-2
2
-1
-10
0
y f(x) = 5(0.6)x (1, 3) 1
© Houghton Mifflin Harcourt Publishing Company
-2
8.3 5 3 1.8
(-1, 8.3)
(2, 27)
(0, 3) 0 1
x
Graph the ordered pairs and connect them with a smooth curve.
20 10 (-1, 1)
f(x) = 5(0.6)
x
(2, 1.8) 2 x
-3 -6
a=3
a=5
y-intercept: (0, 3)
y-intercept: (0, 5)
End Behavior: As x → ∞, y → ∞ and as x → -∞, y → 0.
End Behavior: As x → ∞, y → 0 and as x → -∞, y → ∞.
b=3
Module 14
IN1_MNLESE389762_U6M14L4 686
b = 0.6
686
Lesson 4
4/19/14 11:19 AM
Graphing Exponential Functions
686
11. ƒ(x) = -6(0.7)
AVOID COMMON ERRORS Students may have trouble raising a number to a negative power. Remind them that a number raised to a negative power is the reciprocal of the number raised to the opposite power. For example,
12. ƒ(x) = -4(3)
x
f(x) = -6(0.7)
x -1 0 1 2
()
x
2
6
f(x) = -4(3)
x -1 0 1 2
-8.6 -6 -4.2 -2.9
1 = ___ 1 . 4-2 = __ 4 16
x
x
-1.3 -4 -12 -36
y
y
8
3
4 x
-2
-1
0
x -2 -1 (-1, -1.3)
1 2 (2, -2.9)
-3 f(x) = -6(0.7)x (1, -4.2) -6 (0, -6) (-1, -8.6) -9
a = -4; b = 3
f(x) = 5(2)
x -1 0 1 2
© Houghton Mifflin Harcourt Publishing Company
End Behavior: As x → ∞, y → -∞ and as x → -∞, y → 0.
14. ƒ(x) = -2(0.8)
x
x
21
y
-1 0 1 2
1
x -2
x
-7
0
1 2 (2, -1.3)
a = -2; b = 0.8
y-intercept: (0, -2)
End Behavior: As x → ∞, y → 0 and as x → -∞, y → -∞.
End Behavior: As x → ∞, y → ∞ and as x → -∞, y → 0.
Lesson 14.4
-1
-1 f(x) = -2(0.8)x (1, -1.6) -2 (0, -2) (-1, -2.5) -3
2
y-intercept: (0, 5)
687
y
1
a = 5; b = 2
IN1_MNLESE389762_U6M14L4 687
x
-2.5 -2 -1.6 -1.3
2
-14
Module 14
f(x) = -2(0.8)
(2, 20)
14 f(x) = 5(2)x (1, 10) 7 (-1, 2.5) (0, 5) 0
x
x
2.5 5 10 20
-1
(1, -12)
y-intercept: (0, -4)
End Behavior: As x → ∞, y → 0 and as x → -∞, y → -∞.
-2
1 2 -4 (0, -4) f(x) = -4(3)x -8
-12
a = -6; b = 0.7 y-intercept: (0, -6)
13. ƒ(x) = 5(2)
0
687
Lesson 4
4/19/14 11:19 AM
15. ƒ(x) = 9(3)
16. ƒ(x) = -5(2)
x
f(x) = 9(3)
x -1 0 1 2
x
90
f(x) = -5(2)
x -1 0 1 2
3 9 27 81 y
COGNITIVE STRATEGIES
x
-2.5 -5 -10 -20
(2, 81)
14 7
f(x) = 9(3)x (-1, 3)
30 (0, 9)
(1, 27)
-2 (-1, -2.5)
x
-30 -60
-21
0
1
2
a = 9; b = 3
f(x) = 7(0.4)
f(x) = 6(2)
x -1 0 1 2
17.5 7 2.8 1.1
3 6 12 24
y
y
18
30
12 f(x) = 7(0.4)x (0, 7) 6 (1, 2.8)
20
0
1
(-1, 3)
2
-2
-6
a = 7; b = 0.4
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(2, 24) (1, 12) x
(0, 6)
-1
0
1
2
-10
a = 6; b = 2
y-intercept: (0, 7) End Behavior: As x → ∞, y → 0 and as x → -∞, y → ∞.
f(x) = 6(2)x
10
(2, 1.1) x
x
© Houghton Mifflin Harcourt Publishing Company
-1 0 1 2
Module 14
(2, -20)
x
x
x
-1
asymptote. A horizontal asymptote is a line that the graph gets closer and closer to, but never reaches. The line y = 0 is the horizontal asymptote of the graphs in this lesson.
(1, -10)
End Behavior: As x → ∞, y → -∞ and as x → -∞, y → 0. 18. ƒ(x) = 6(2)
x
-2
2
a = -5; b = 2
End Behavior: As x → ∞, y → ∞ and as x → -∞, y → 0.
(-1, 17.5)
x 1 (0, -5)
y-intercept: (0, -5)
y-intercept: (0, 9)
17. ƒ(x) = 7(0.4)
0
-7 f(x) = -5(2)x -14
-1
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Introduce students to the term horizontal
y
60
-2
Show students that the graph of y = -ab x is the reflection of y = ab x over the x-axis.
x
y-intercept: (0, 6)
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End Behavior: As x → ∞, y → ∞ and as x → -∞, y → 0.
Lesson 4
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Graphing Exponential Functions
688
19. Identify the domain and range of each function. Make sure to provide these answers using inequalities. ⎫ ⎫ ⎧ ⎧ x Domain: ⎨x⎢ -∞ < x < ∞⎬, Range: ⎨ y⎢y > 0⎬ a. ƒ(x) = 3(2) ⎭ ⎭ ⎩ ⎩ x ⎫ ⎫ ⎧ ⎧ Domain: ⎨x⎢ -∞ < x < ∞⎬, Range: ⎨ y⎢y > 0⎬ b. ƒ(x) = 7(0.4) ⎭ ⎭ ⎩ ⎩ x ⎫ ⎫ ⎧ ⎧ c. ƒ(x) = -2(0.6) Domain: ⎨x⎢ -∞ < x < ∞⎬, Range: ⎨ y⎢y < 0⎬ ⎭ ⎭ ⎩ ⎩ x ⎫ ⎫ d. ƒ(x) = -3(4) ⎧ ⎧ Domain: ⎨x⎢ -∞ < x < ∞⎬, Range: ⎨ y⎢y < 0⎬ ⎭ ⎭ ⎩ ⎩ x e. ƒ(x) = 2(22) ⎫ ⎫ ⎧ ⎧ Domain: ⎨x⎢ -∞ < x < ∞⎬, Range: ⎨ y⎢y > 0⎬ ⎭ ⎭ ⎩ ⎩
JOURNAL Have students describe the four basic shapes, including the quadrants each graph occupies and the end behavior of each graph. Also have them write an exponential function of each type.
20. Statistics In 2000, the population of Massachusetts was 6.3 million people and was growing at a rate of about 0.32% per year. At this growth rate, the function x ƒ(x) = 6.3(1.0032) gives the population, in millions x years after 2000. Using this model, find the year when the population reaches 7 million people.
f(x) = 6.3(1.0032) 7 = 6.3(1.0032)
1.11 ≈ 1.0032
x x
x
In(1.11) ≈ x In(1.0032) 0.104 ≈ 0.003x
x ≈ 34.7 years
© Houghton Mifflin Harcourt Publishing Company • Image Credits: © Spirit of America/Shutterstock
The population will reach approximately 7 million people during the year 2034. 21. Physics A ball is rolling down a slope and continuously picks up speed. Suppose x the function ƒ(x) = 1.2(1.11) describes the speed of the ball in inches per minute. How fast will the ball be rolling in 20 minutes? Round the answer to the nearest whole number.
f(x) = 1.2(1.11) f(x) = 1.2(1.11)
x 20
f(x) ≈ 1.2(8.06)
f(x) ≈ 9.67
The ball will be rolling at a rate of about 10 inches per minute after 20 minutes. H.O.T. Focus on Higher Order Thinking
22. Draw Conclusions Assume that the domain of the function ƒ(x) = 3(2) is the set of all real numbers. What is the range of the function? x
The range of the function is positive all real numbers since f(x) is always positive. 23. What If? If b = 1 in an exponential function, what will the graph of b look like?
The graph of b will be a horizontal line at y = a.
24. Critical Thinking Using the graph of an exponential function, how can b be found?
Identify two points (x 1, y 1) and (x 2, y 2) on the graph of the exponential ― x - x y2 y2 __ . function with x 1 < x 2. If x 2 - x 1 then b = __ y 1 , otherwise b = y1 2
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25. Critical Thinking Use the table to write the equation for the exponential function.
x
f(x)
-1
4 _ 5
0
4
1
20
2
100
INTEGRATE TECHNOLOGY
The y-intercept of the function is located at (0, 4), so a = 4. To find b,
Have students use a graphing calculator to graph both equations on the same graph. Compare the two graphs.
divide the y-coordinate of (1, 20) by the y-coordinate of (0, 4). 20 b = __ =5 4
f(x) = 4(5)
x
INTEGRATED MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Have students describe how the pumpkin
Lesson Performance Task A pumpkin is being grown for a contest at the state fair. Its growth can be modeled by the n equation P = 25(1.56) , where P is the weight of the pumpkin in pounds and n is the number of weeks the pumpkin has been growing. How much does the pumpkin weigh after 5 weeks? By what percentage does the pumpkin grow every week? After how many weeks will the pumpkin be 80 pounds?
grew over the period of time until it reached 150 pounds. They could say that the pumpkin grew to a weight of 80 pounds in about 2.6 weeks, while it took about 4.9 weeks to gain another 70 pounds. They could also say that the rate provided in the first equation is 156% per week, while the second equation shows the rate declined to 123%.
After the pumpkin grows to 80 pounds, its growth is reduced to P = 25(1.23) . Estimate when the pumpkin will reach 150 pounds. n
P = 25(1.56) ≈ 230.97 pounds 5
The pumpkin grows by 56% every week. 80 = 25(1.56)
n
3.2 = (1.56)
n
In(3.2) = n • In(1.56) 1.16 ≈ .44n 2.6 ≈ n
The pumpkin will exceed 80 pounds after around 2.8 weeks. © Houghton Mifflin Harcourt Publishing Company
It only needs to grow 150 - 80 = 70 more pounds. Let P = 70 and solve the new equation. Add the solution to this equation to the original estimate to find the total time it will take to grow to 150 pounds. 70 = 25(1.23)
n
2.8 = (1.23)
n
In(2.8) = n • In(1.23) 1.03 ≈ .21n 4.9 ≈ n
4.9 + 2.6 = 7.5 It will take the pumpkin about 7.5 weeks to grow to 150 pounds.
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Have students research how the weight of a giant pumpkin can be estimated before it is picked. Have students do an Internet search for “over the top (OTT) weight tables.” There are three measurements that must be found and added together to get the over-the-top measurement. Have students use the data they find to determine what the relationship appears to be between the over-the-top measurement in inches and the weight of a giant pumpkin in pounds. A graph of these values is a scatter plot that could be represented by an exponential function.
4/19/14 11:19 AM
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Graphing Exponential Functions
690
LESSON
14.5
Name
Transforming Exponential Functions
Class
Date
14.5 Transforming Exponential Functions Essential Question: How does the graph of f(x) = ab x change when a and b are changed?
Common Core Math Standards The student is expected to: COMMON CORE
Explore
F-BF.B.3
Changing the Value of b in f(x) = b x
Investigate the effect of b on the function ƒ(x) = b x.
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Also F-IF.C.9, F-BF.A.1b
Complete the table of values for the functions ƒ 1(x) = 1.2 x and ƒ 2(x) = 1.5 x. Use a calculator to find the values and round to the nearest thousandth if necessary.
f 1(x) = 1.2 x
f 2(x) = 1.5 x
0.694
0.444
–1
0.833
0.667
0
1
1
1
1.2
1.5
2
1.44
x –2
Mathematical Practices COMMON CORE
Resource Locker
MP.7 Using Structure
Language Objective
Describe how the graph of an exponential function changes when you add a constant to the function.
2.25
Select the option that makes the statement true. ƒ 1(x) | ƒ 2(x) increases more quickly as x increases. ƒ 1(x) | ƒ 2(x) approaches 0 more quickly as x decreases.
If a > 0 and b > 1, increasing the value of b makes the graph rise more quickly as x increases. For a > 0 and 0 < b < 1, increasing the value of b makes the graph fall more gradually as x increases. Increasing the absolute value of a stretches the graph vertically, and changing the sign of a reflects the graph across the x-axis.
PREVIEW: LESSON PERFORMANCE TASK
The y-intercept of ƒ 1(x) is 1 . The y-intercept of ƒ 2(x) is 1 .
Fill in the table of values for the functions ƒ 3(x) = 0.6 x and ƒ 4(x) = 0.9 x. Round to the nearest thousandths again.
x –2 –1
f 3(x) = 0.6 x
f 4(x) = 0.9 x
2.778
1.235
1.667
1.111
0
1
1
1
0.6
0.9
2
0.36
0.81
ƒ 3(x) | ƒ 4(x) increases more quickly as x decreases. ƒ 3(x) | ƒ 4(x) approaches 0 more quickly as x increases.
The y-intercept of ƒ 3(x) is 1 . The y-intercept of ƒ 4(x) is 1 .
Module 14
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gh "File info"
made throu
be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
Date Class
ponential
forming Ex 14.5 Transtions Func
Name
changed? and b are e when a x ab chang f(x + k) for of f(x) = f(kx), and the graph + k, kf(x), How does f(x) by f(x) Question: of replacing , F-BF.A.1b Essential on the graph x Also F-IF.C.9 y the effect negative)... =b COMMON F-BF.B.3 Identif of k (both positive and CORE b in f(x) of e values c specifi the Valu
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HARDCOVER PAGES 691702
g
Changin
Explore
b n ƒ(x) = b on the functio
x
.
a calculator
x Use . x of ƒ 2(x) = 1.5 the effect = 1.2 and ons ƒ 1(x) if necessary. for the functi of values t thousandth the table the neares Complete round to x values and to find the x f 2(x) = 1.5 f 1(x) = 1.2 0.444 x 0.694 0.667 –2 0.833 1 –1 1 1.5 0 1.2 2.25 1 1.44 2 true. statement makes the option that x increases. Select the quickly as ses more increa ses. ) x ( ƒ 1(x) | ƒ 2 y as x decrea 0 more quickl . ) approaches x ( ƒ 2 | (x) is 1 ) x ƒ 1( rcept of ƒ 2 x Round to the 1 . The y-inte . (x) is x ƒ 4(x) = 0.9 rcept of ƒ 1 y-inte = 0.6 and The ons ƒ 3(x) for the functi of values table the Fill in ndths again. x nearest thousa x f 4(x) = 0.9 f 3(x) = 0.6 1.235 x 2.778 1.111 –2 1.667 1 –1 1 0.9 0 0.6 0.81 1 0.36 2 x decreases. quickly as ses more (x) increa x increases. ƒ 3(x) | ƒ 4 quickly as . aches 0 more (x) is 1 (x) appro rcept of ƒ 4 ƒ 3(x) | ƒ 4 . The y-inte (x) is 1 rcept of ƒ 3 The y-inte
Investigate
Resource Locker
Watch for the hardcover student edition page numbers for this lesson.
© Houghto
n Mifflin
Harcour t
Publishin
View the Engage section online. Discuss how heat flows from a hotter object to a colder object until both are the same temperature. Then preview the Lesson Performance Task.
y g Compan
Essential Question: How does the graph of f(x) = ab x change when a and b are changed?
© Houghton Mifflin Harcourt Publishing Company
ENGAGE
Lesson 5
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Lesson 14.5
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Reflect
1.
EXPLORE
Consider the function, y = 1.3 x. How will its graph compare with the graphs of ƒ 1(x) and ƒ 2(x)? Discuss end behavior and the y-intercept. All three graphs have the same y-intercept of 1. The graph of y = 1.3 x falls between
Changing the Value of b in f(x) = b x
the other two graphs and increases more quickly than f 1(x) but less quickly than f 2(x) as x increases to the right of 0. The graph falls more quickly than that of f 1(x) but less
INTEGRATE TECHNOLOGY
quickly than that of f 2(x) as x decreases to the left of 0.
After students have used tables to explore the effects of changing the value of b in the function f(x) = b x, have students use calculators to graph functions with different values of b on the same grid.
Changing the Value of a in f(x) = ab with b > 1 x
Explain 1
Multiplying a growing exponential function (b > 1) by a constant a does not change the growth rate, but it does stretch or compress the graph vertically, and reflects the graph across the x-axis if a < 0.
A vertical stretch of a graph is a transformation that pulls the graph away from the x-axis. By multiplying the y-value of each (x, y) pair by a, where ⎜a⎟ > 1, the graph is stretched by a factor of ⎜a⎟.
QUESTIONING STRATEGIES
A vertical compression of a graph is a transformation that pushes the graph toward the x-axis. By multiplying the y-value of each (x, y) pair by a, where ⎜a⎟ < 1, the graph is compressed
⎜⎟
For two values of b that are both greater than 1, why is the graph of y = b x steeper for the larger value of b? The value of b determines by what factor the function grows for each unit increase in the value of x. A larger value of b means more growth per unit and a steeper graph.
by a factor of __a1 .
Make a table of values for the function given. Then graph it on the same coordinate plane with the graph of y = 1.5 x. Describe the end behavior and find the y-intercept of each graph.
Example 1
ƒ(x) = 0.3(1.5)
x –2
x
f(x) = 0.3(1.5)
x
4
0.133 0.2
0
0.3
1
0.45
2
0.675
3
1.013
4
1.519
2 x -4
-2
0
2
EXPLAIN 1
© Houghton Mifflin Harcourt Publishing Company
–1
y
4
-2 -4
End Behavior: ƒ(x) → ∞ as x → ∞ ƒ(x) → 0 as x → -∞
Changing the Value of a in f(x) = ab x with b > 1 INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Select a value for a and b, then graph both
y-intercept: 0.3
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f(x) = abx and f(x) = –ab x. Discuss with students why the graph of f(x) = ab x is a reflection of f(x) = –ab x over the x-axis.
Lesson 5
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Integrate Mathematical Practices
This lesson provides an opportunity to address Mathematical Practice MP.7, which calls for students to “look for and make use of structure.” Students will compare exponential functions. They will explore how changing the parameters of the functions affects the shapes of their graphs, including how quickly the graphs rise or fall, end behavior, and y-intercepts. They will identify patterns that will allow them to predict how increasing, decreasing, or changing the sign of a parameter will affect the graph of an exponential function.
4/19/14 11:30 AM
QUESTIONING STRATEGIES What are the domain and range of an exponential function f(x) = ab x when a is positive? The domain will be all real numbers, and the range will be all numbers greater than 0. What are the domain and range of an exponential function f(x) = ab x when a is negative? The domain will be all real numbers, and the range will be all numbers less than 0.
Transforming Exponential Functions 692
ƒ(x) = -2(1.5)
B
x
f(x) = -2(1.5)
x
x
4
-0.395
–4
2
-0.593
–3
x
-0.889
–2
-4
-1.333
–1
-2
-2
0
0
2
4
-2 -4
-3
1
y
-4.5
2
End Behavior: ƒ(x) → -∞ as x → ∞ ƒ(x) →
0 as x → -∞
y-intercept: -2 Reflect
2.
Discussion What can you say about the common behavior of graphs of the form ƒ(x) = ab x with b > 1? What is different when a changes sign? All graphs of the form f(x) = ab x with b > 1 approach 0 as x approaches -∞ and have a y-intercept at (0, a). The sign of a determines the end behavior as x approaches ∞; for a > 0, f(x) increases toward infinity, and for a < 0, f(x) decreases toward negative infinity.
Your Turn
© Houghton Mifflin Harcourt Publishing Company
Graph each function, and describe the end behavior and find the y-intercept of each graph. 3.
ƒ(x) = -0.5(1.5)
x
4
4.
ƒ(x) = 4(1.5)
x
y
y 6
2
4
x -4
-2
0
2
4
2
-2
x -4
-4
-2
0
End behavior:
End behavior:
f(x) → -∞ as x → ∞
f(x) → ∞ as x → ∞
y-intercept: -0.5
y-intercept: 4
f(x) → 0 as x → -∞
Module 14
2
4
f(x) → 0 as x → -∞
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Whole Class Activity
As a class, work together to determine what occurs when a constant h is introduced in the exponential function f(x) = b x – h. Have students select a value for b and three values for h, then write three functions in the form f(x) = b x – h using their chosen values. Create a table that shows the output of each function for several x-values. Graph all three functions on the same coordinate grid. Discuss how h affects the parent function f(x) = b x. Students should realize that positive values of h translate the graph to the right, while negative values of h translate the graph to the left.
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Changing the Value of a in f(x) = ab x with 0 < b < 1
Explain 2
EXPLAIN 2
Multiplying a decaying exponential function (b < 1) by a constant a does not change the growth rate, but it does stretch or compress the graph vertically.
Changing the Value of a in f(x) = ab x with 0 < b < 1
Make a table of values for the function given. Then graph it on the same coordinate plane with the graph of y = 0.6 x. Describe the end behavior and find the y-intercept of each graph.
Example 2
ƒ(x) = -3(0.6)
x
4
f(x) = –3(0.6) x
x –1
–3
1
–1.8
2
–1.08
3
–0.648
Remind students that variables can have different meanings when they are used in different functions. Students should understand that while b in a linear function y = mx + b represents the y-intercept, b in an exponential function y = ab x does not represent the y-intercept.
2
–5
0
AVOID COMMON ERRORS
y
x -4
-2
0 -2
End behavior: ƒ(x) → 0 as x → ∞
ƒ(x) → ∞ as x → ∞
QUESTIONING STRATEGIES
y-intercept: -3
ƒ(x) = 0.5 (0.6)
x
4
f(x) = 0.5(0.6) 3.858
–3
2.315
–2
1.389
–1
0.833
0
0.5
1
0.3
2
0.18
2 x -4
-2
0
2
4
-2 -4
End Behavior: ƒ(x) →
0
as x → ∞
ƒ(x) → ∞ as x → -∞ y-intercept: 0.5 Reflect
5.
Discussion What can you say about the common behavior of graphs of the form ƒ(x) = ab x with 0 < b < 1? What is different when a changes sign? All graphs of the form f(x) = ab x with 0 < b < 1 approach 0 as x approaches ∞ and have a
© Houghton Mifflin Harcourt Publishing Company
x –4
How can the values of a and b tell you in which direction the graph of an exponential function goes as x increases, and in which quadrants the graph is located? When b > 1, the graph moves away from the x-axis as x increases. When 0 < b < 1, the graph approaches the x-axis as x increases. When a > 0, the graph is in quadrants 1 and 2 (above the x-axis). When a < 0, the graph is in quadrants 3 and 4 (below the x-axis).
y
x
How is the y-intercept of a function f(x) = ab x related to the value of a? Explain. Since the y-intercept of a function is its value when x = 0, and b 0 = 1 for any value of b, the y-intercept of f(x) = a b x is always equal to a.
y-intercept at (0, a). The sign of a determines the end behavior as x approaches -∞;
for a > 0, f(x) approaches ∞ as x approaches -∞, and for a < 0, f(x) approaches -∞ as x approaches -∞. Module 14
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Visual Cues
4/19/14 11:30 AM
How is the effect of changing a when 0 < b < 1 similar to the effect of changing a when b > 1? In both cases, increasing the absolute value of a creates a vertical stretch of the graph and decreasing the absolute value of a creates a vertical compression of the graph. Changing the sign of a reflects the graph across the x-axis.
Have students graph, on the same grid, a parent exponential function and exponential functions in which the value of a or b has been changed or to which a constant has been added. Students can draw each graph in a different color and write the equation for each function in the same color as its graph. Then have them circle the value that was changed in each equation. Students may wish to keep this graph as a visual reference to use when working with transformations of other exponential functions.
Transforming Exponential Functions
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Your Turn
EXPLAIN 3
Graph each function, and describe its end behavior and y-intercept. 6.
Adding a Constant to an Exponential Function
ƒ(x) = 2(0.6)
x
7.
ƒ(x) = -0.25(0.6)
y
4
6
-2
x
constant to a parent exponential function shifts the graph up or down but does not change the shape of the graph. Contrast this to the effect of either changing the base b in the function or multiplying the function by a constant a, both of which do change the shape of the graph.
-2
0
2
4
0
2
4
-2 -4
End behavior:
End behavior:
f(x) → 0 as x → ∞
f(x) → 0 as x → ∞
y-intercept: 2
y-intercept: -0.25
f(x) → -∞ as x → -∞
f(x) → ∞ as x → -∞
Adding a Constant to an Exponential Function
Explain 3
Adding a constant to an exponential function causes the graph of the function to translate up or down, depending on the sign of the constant. Example 3
QUESTIONING STRATEGIES
Make a table of values for each function and graph them together on the same coordinate plane. Find the y-intercepts, and explain how they relate to the translation of the graph.
ƒ(x) = 2 x and g(x) = 2 x + 2
© Houghton Mifflin Harcourt Publishing Company
Can the graph of an exponential function be translated in any desired direction by adding a constant to the function? Explain. No; adding a constant to an exponential function can only translate the graph vertically, either up or down. It cannot translate the graph horizontally.
x -4
2 -4
y
2
4
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Make sure students understand that adding a
x
x
f(x) = 2 x
g(x) = 2 x + 2
8
-2
0.25
2.25
6
-1
0.5
2.5
0
1
3
1
2
4
2
4
6
y
4 2 x -2
0
2
4
The y-intercept of ƒ(x) is 1. The y-intercept of g(x) is 3. The y-intercept of g(x) is 2 more than that of ƒ(x) because g(x) is a vertical translation of ƒ(x) up by 2 units.
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Communicate Math To accurately describe the transformations of functions explored in this lesson, students must be careful to use precise language. Provide sentence starters to help English learners frame explanations of changes in the behavior of a function. For example: • If [b > 1 / b < 1], increasing the value of b causes the function to [approach infinity / approach zero] more [quickly / slowly] as x increases. • [Increasing / Decreasing] the absolute value of a causes a [vertical stretch / vertical compression / translation / reflection] of the graph.
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B
ƒ(x) = 0.7 x and g(x) = 0.7 x - 3
g(x) = 0.7 x - 3
x
f(x) = 0.7 x
-2
2.041
-1
1.429
0
1.0
1
0.7
-2.3
-2
2
0.49
-2.51
-4
The y-intercept of f(x) is
4
-0.959
2
-1.571
x
-2
1
y
-4
-2
0
2
4
.
The y-intercept of g(x) is -2 . The y-intercept of g(x) is 3 more | less than that of ƒ(x) because g(x) is a vertical translation of ƒ(x) up | down by 3 units. Reflect
8.
What do you think will happen to the y-intercept of an exponential function with both a stretch and a x translation, such as ƒ(x) = 3(0.7) + 2? The y-intercept is stretched up from 1 to 3 by the vertical stretch and then translated up
from 3 to 5 by the translation. Your Turn
Graph the functions together on the same coordinate plane. Find the y-intercepts, and explain how they relate to the translation of the graph. 9.
10. ƒ(x) = 2(1.5) and g(x) = 2(1.5) - 3 x
ƒ(x) = 0.4 x and g(x) = 0.4 x + 4 y
x
y
2
4
x -4
2 x -4
-2
0
© Houghton Mifflin Harcourt Publishing Company
4
6
2
4
-2
0
2
4
-2 -4
The y-intercept of f(x) is 1.
The y-intercept of f(x) is 2.
The y-intercept of g(x) is 3 more than that of f(x) because g(x) is a vertical translation of f(x) up by 3 units.
The y-intercept of g(x) is 3 less than that of f(x) because g(x) is a vertical translation of f(x) down by 3 units.
The y-intercept of g(x) is 4.
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The y-intercept of g(x) is -1.
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Transforming Exponential Functions
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Elaborate
ELABORATE
11. How do you determine the y-intercept of an exponential function ƒ(x) = ab x + k that has been both stretched and translated? The y-intercept of all the parent exponential functions f(x) = b x is 1. First multiply 1 by a to
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Some students may feel that the change that
find the effect of the stretch, and then add k to find the effect of the translation. 12. Describe the end behavior of a translated exponential function ƒ(x) = b x + k as x approaches -∞.
Since all points are shifted by k, the function approaches k as x approaches -∞.
13. Essential Question Check-in If a and b are positive real numbers and b ≠ 1, how does the graph of ƒ(x) = (ab x) change when b is changed? If b > 1, increasing b makes the graph rise more quickly as x increases. If 0 < b < 1,
results from adding a constant to an exponential function is not a translation of the graph, because the curves can appear to be closer to each other on one side of the grid than on the other side. To illustrate that the transformation represents a translation, have students determine the vertical distance between the transformed graph and the graph of its parent function for any x-value. They should find that the distance is the same for all values of x, verifying that both graphs are increasing or decreasing at the same rate.
increasing b makes the graph fall more gradually as x increases.
Evaluate: Homework and Practice • Online Homework • Hints and Help • Extra Practice
Exercises 1 and 2 refer to the functions ƒ 1 (x) = 2.5 x and ƒ 2 (x) = 3 x. 1.
Which function grows faster as x increases toward ∞ ?
f 2 (x) 2.
SUMMARIZE THE LESSON
f 2 (x)
Exercises 3 and 4 refer to the functions ƒ 1 (x) = 0.5 x and ƒ 2 (x) = 0.7 x. © Houghton Mifflin Harcourt Publishing Company
How does changing a, b, or k change the graph of an exponential function in the form f(x) = ab x + k? Increasing the absolute value of a stretches the graph vertically; decreasing the absolute value of a compresses the graph vertically. If b > 1, increasing the value of b makes the graph rise more quickly as x increases. If 0 < b < 1, increasing the value of b makes the graph approach the x-axis more slowly as x increases. Adding a constant k translates the graph of the parent function upward if k > 0, and translates the graph downward if k < 0.
Which function approaches 0 faster as x decreases toward -∞ ?
3.
Which function grows faster as x decreases toward -∞ ?
f 1 (x) 4.
Which function approaches 0 faster as x increases toward ∞ ?
f 1 (x) Label each of the following functions, g (x), as a vertical stretch or a vertical compression of the parent function, ƒ (x), and tell whether it is reflected about the x-axis. 5.
g (x) = 0.7(0.5) , ƒ (x) = 0.5 x x
vertical compression, no reflection
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Exercise
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Lesson 14.5
6.
g (x) = -1.2(5) , ƒ (x) = 5 x x
vertical stretch and reflection
Lesson 5
697
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1–8
1 Recall of Information
MP.7 Using Structure
9–12
1 Recall of Information
MP.6 Precision
13–16
1 Recall of Information
MP.7 Using Structure
17
2 Skills/Concepts
MP.4 Modeling
18
2 Skills/Concepts
MP.7 Using Structure
19–20
2 Skills/Concepts
MP.4 Modeling
4/19/14 11:30 AM
Label each of the following functions, g (x), as a vertical stretch or a vertical compression of the parent function, ƒ(x), and tell whether it is reflected about the x-axis. 7.
EVALUATE
8.
y
y
f(x)
2
6
f(x)
1 x -2
-1
0 -1
g(x)
2
1 g(x)
x -4
-2
0
2
ASSIGNMENT GUIDE
4
vertical stretch, no reflection
vertical compression and reflection
Find the y-intercept for each of the functions, g (x), from Exercises 5–8. 9.
g (x) = 0.7(0.5)
10. g (x) = -1.2(5)
x
x
-1.2
0.7 11. Use g (x) from Exercise 7.
12. Use g (x) from Exercise 8.
-0.5
3
Describe the translation of each of the functions, g (x), compared to the parent function, ƒ(x).
14. ƒ(x) = -2(1.5) , g (x) = -2(1.5) - 2 x
13. ƒ (x) = 0.4 x, g (x) = 0.4 x + 5
g (x) is translated up by 5 units from f (x). 15.
16.
-2
0
3 g(x) 2
x 4
Module 14
g(x)
1
-2
x -2
g (x) is translated down by 3 units from f (x).
Exercise
2
f(x)
-4
IN1_MNLESE389762_U6M14L5 698
y
-1
0
2
3
g (x) is translated up by 3 units from f (x).
Exercises 1–4, 27–29
Example 1 Changing the Value of a in f(x) = ab x with b > 1
Exercises 6–7, 10–11, 20–25
Example 2 Changing the Value of a in f(x) = ab x with 0 < b < 1
Exercises 5, 8–9, 12, 17–18, 26
Example 3 Adding a Constant to an Exponential Function
Exercises 13–16, 19
students might use the word slope to describe either the parameter a or b. Remind students that slope refers to the constant rate of change found in a linear function. An exponential function does not have a constant rate of change.
Lesson 5
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Depth of Knowledge (D.O.K.)
1
-1
COMMON CORE
Mathematical Practices
1 Recall of Information
MP.6 Precision
23
2 Skills/Concepts
MP.6 Precision
24
1 Recall of Information
MP.6 Precision
25–26
2 Skills/Concepts
MP.6 Precision
27–29
3 Strategic Thinking
MP.3 Logic
21–22
Explore Changing the Value of b in f(x) = b x
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 When working with exponential functions,
© Houghton Mifflin Harcourt Publishing Company
-4
4
f(x)
2
Practice
g (x) is translated down by 2 units from f (x).
y 4
x
Concepts and Skills
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Transforming Exponential Functions
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The height of the nth bounce of a bouncy ball dropped from a height of 10 feet can be characterized by a decaying n exponential function, h(n) = 10(0.8) , where each bounce reaches 80% of the height of the previous bounce.
TECHNOLOGY When using calculators to graph exponential functions, remind students that a calculator will read numbers separated by parentheses as implied multiplication, so a x function such as f(x) = 3(4) can be entered either as y = 3*4^x or as y = 3(4)^x.
17. Write the new function if the ball is dropped from 5 feet.
h (n) = 5(0.8)
n
18. What kind of transformation was that from the original n function, h (n) = 10(0.8) ?
a vertical compression
AVOID COMMON ERRORS Students may have trouble differentiating between a vertical stretch or compression and a vertical translation. Remind them that changing the value of a in f(x) = ab x causes a vertical stretch or compression, which is a change in the shape of the graph. Adding a constant k produces a vertical translation, which slides the graph up or down without changing its shape.
19. Write the function that describes what happens if the ball is dropped from 10 feet above a table top that is at a height of 3 feet. The function is translated up by 3 feet.
k = 3 h (n) = 10(0.8) + 3 n
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Rubberball/Erik Isakson/Getty Images
20. Biology Unrestrained growth of cells in a petri dish can be extremely rapid, with a single cell growing into a number of cells, N, given by the formula, N (t) = 8 t , after t hours. a. Write the formula for the number of cells in the petri dish when a culture is started with 50 isolated cells.
N (t) = 50(8)
b. How many cells do you expect after 3 hours? 3 N (3) = 50(8) = 50 ∙ 512 = 25,600 cells
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Lesson 14.5
t
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Lesson 5
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A bank account with an initial deposit of $1000 and an interest rate of 5% increases by 5% each year. The balance (B) as a function of time in years (t) can t be described by an exponential function: B(t) = 1000(1.05)
MODELING When working with exponential functions that have an added constant, draw the graph of a parent exponential function on a grid on a clear transparency sheet. On another clear transparency sheet, trace the graph only. Slide the second sheet over the first sheet to show a translation. Visual learners may wish to make and use their own pairs of transparency sheets to help with translations.
21. What parameter of the exponential form ƒ (x) = ab x + k represents the initial balance of $1000?
a 22. What is the y-intercept of B (t)?
1000 23. What parameter would change if the interest rate were changed to 7%?
b 24. Which bank account balance grows faster, the one with 5% interest or the one with 7% interest?
the one with the 7% interest rate 25. What kind of transformation is represented by changing the initial balance to $500?
a vertical compression 26. Match the graph to the characteristics of the function ƒ(x) = ab x. 1.
a < 0, b > 1
b
2.
a > 0, b < 1
c
3.
a > 1, b > 1
4.
a
a < 0, b < 1 d
y
y
x
x
a.
b. © Houghton Mifflin Harcourt Publishing Company
y
y x x
c.
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d.
700
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Transforming Exponential Functions
700
JOURNAL
H.O.T. Focus on Higher Order Thinking
27. Critical Thinking Describe how the graph of ƒ (x) = ab x changes for a given positive value of a as you increase the value of b when b > 1. Discuss the rise and fall of the graph and the y-intercept.
Have students create a graphic organizer like the one shown to explain each type of transformation studied in this lesson. Students may include either verbal explanations or graphs to show the effect of each change.
The y-intercept equals a for all values of b > 1 because ab 0 = a. As the value of b increases, the graph rises more quickly as x increases to the right of 0, and it falls more quickly as x decreases to the left of 0.
f(x) = ab x
Changing the absolute value of a:
Changing the Adding a value of b when positive b > 1: constant k:
Changing the sign of a:
Changing the Adding a value of b when negative 0 < b < 1: constant k:
28. Communicate Mathematical Ideas Consider the functions ƒ 1 (x) = (1.02) x and ƒ 2 (x) = (1.03) . Which function increases more quickly as x increases to the right of 0? How do the growth factors support your answer? x
f 2 (x) grows more quickly. f 2 (x) has the greater growth rate (3% rather than 2%), so you would expect f 2 (x) to increase more quickly as x increases to the right of 0.
29. Communicate Mathematical Ideas Consider the function ƒ 1(x) = (0.94) x and ƒ 2 (x) = (0.98) . Which function decreases more quickly as x increases to the right of 0? How do the growth factors support your answer? x
© Houghton Mifflin Harcourt Publishing Company
f 1 (x) decreases more quickly. f 1 (x) has the greater decay rate (6% rather than 2%), so you would expect f 1 (x) to decrease more quickly as x increases to the right of 0.
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Lesson Performance Task
QUESTIONING STRATEGIES
A coffee shop serves two patrons cups of coffee. The initial temperature of the coffee is 170 °F. As the coffee sits in the 70 °F room, the temperature follows the pattern of a transformed exponential function. One patron leaves her coffee untouched, resulting in a slow cooling toward room temperature. The other patron is in a hurry and stirs her coffee, resulting in a faster cooling rate.
In the Lesson Performance Task, is the temperature of the coffee described by an increasing or decreasing exponential function? What can you conclude about the value of b? decreasing; b < 1
Both cups of coffee can be modeled with transformed exponential functions of the form T (t) = ab t + k.
What value does the temperature of the coffee approach as t approaches infinity? In the general exponential function T(t) = ab t + k with b < 1, what value does ab t approach as t approaches infinity? How can you use this information to find one of the parameters of the temperature functions? As t approaches infinity, the temperature of the coffee approaches room temperature, 70°F, and ab t approaches 0. Therefore, as t approaches infinity, the equation T(t) = ab t + k becomes 70 = 0 + k, so k = 70 for both functions.
Each minute, the unstirred coffee gets 10% closer to room temperature, and the stirred coffee gets 20% closer. Find the functions T s (t) and T u (t) for the stirred and unstirred cups of coffee, fill in the table of values, and graph the functions. Determine how long it takes each cup to drop below 130 °F (don’t try to solve the equations exactly, just use the table to answer to the nearest minute).
Time (minutes)
Temperature (°F, unstirred)
Temperature (°F, stirred)
0
170.0
170.0
1
160.0
150.0
2
151.0
134.0
3
142.9
121.0
4
135.6
111.0
5
129.0
102.8
T
160 140
You can also determine that b < 1 and a > 0 for both cups of coffee since they are both decreasing functions that approach a constant value as time increases.
unstirred
120 100
To find a, use the y-intercept. In this case, the temperature of the coffee at t = 0 , or 170 °F:
stirred
80
t 0
INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Have students graph the two
First, recognize that the end behavior as described is for T s (t) and T u (t) to approach 70 °F as time increases, so k = 70 for both functions.
2
4 6 8 Time (minutes)
10
y-intercept = a + k, so 170 = a + 70 and 100 = a for both functions.
A 10% drop each minute for the unstirred coffee means b = (100% - 10%) = 90% = 0.9. For the stirred coffee, b = (100% - 20%) = 80% = 0.8.
© Houghton Mifflin Harcourt Publishing Company
Temperature (°F)
180
temperature functions on a graphing calculator to explore the temperature change over a longer period of time. Have them adjust the viewing window so that the maximum and minimum values of y represent the initial and final temperatures (170°F and 70°F), respectively. Then have them use the TRACE feature to determine how long it takes for the coffee temperature to drop below 71°F. Students may find that the stirred cup takes about 21 minutes and the unstirred cup takes about 44 minutes.
T u (t) = 100(0.9) + 70 for the unstirred coffee t
T s (t) = 100(0.8) + 70 for the stirred coffee t
The unstirred cup takes 5 minutes to drop below 130 °F, and the stirred cup takes 3 minutes to drop below 130 °F. Module 14
702
Lesson 5
EXTENSION ACTIVITY IN1_MNLESE389762_U6M14L5 702
Have students research Newton’s law of cooling and use it to explain why an exponential function may be used to model the cooling of a cup of coffee. Newton’s law of cooling states that the rate of heat loss of an object, or its temperature change per unit time, is proportional to the difference in temperature between the object and its surroundings. As the coffee cools, the temperature difference between the coffee and its surroundings decreases, so the temperature change per minute also decreases. In other words, the temperature decreases at a steadily decreasing rate, as described by an exponential function.
4/19/14 11:29 AM
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Transforming Exponential Functions
702
MODULE
14
STUDY GUIDE REVIEW
Geometric Sequences and Exponential Functions
Study Guide Review
Essential Question: How can you use geometric sequences and exponential functions to solve real-world problems?
ASSESSMENT AND INTERVENTION
KEY EXAMPLE
(Lesson 14.1)
Find the common ratio r for the geometric sequence 2, 6, 18, 54, … and use r to find the next three terms. 6 = 3, so the common ratio r is 3. _ 2 For this sequence, ƒ(1) = 2, ƒ(2) = 6, ƒ(3) = 18, and ƒ(4) = 54.
ƒ(4) = 54, so ƒ(5) = 54(3) = 162.
Assign or customize module reviews.
MODULE
14
Key Vocabulary
common ratio (razón común) explicit rule (fórmula explícita) exponential function (función exponencial) geometric sequence (sucesión geométrica) recursive rule (fórmula recurrente)
ƒ(5) = 162, so ƒ(6) = 162(3) = 486.
ƒ(6) = 486, so ƒ(7) = 486(3) = 1458. The next three terms of the sequence are 162, 486, and 1458.
MODULE PERFORMANCE TASK
KEY EXAMPLE
(Lesson 14.3)
Write an equation for the exponential function that includes the points (2, 8) and (3, 16). 16 Find b by dividing the function value of the second pair by the function value of the first: b = __ = 2. 8
COMMON CORE
Evaluate the function for x = 2 and solve for a.
Mathematical Practices: MP.1, MP.2, MP.4, MP.6, MP.8 F-LE.A.2, F-BF.A.1a
ƒ(x) = ab x
ƒ(x) = a ⋅ 2 x 8 = a ⋅ 22
SUPPORTING STUDENT REASONING
• What is the maximum number of days that can be used? Students should assume that the required time is no more than 6 or 7 days. • How long does a round of sharing take? Students can assume that each round of sharing takes a full day.
© Houghton Mifflin Harcourt Publishing Company
Students should begin this problem by listing appropriate strategies and choosing one. Here are some questions they might have.
8=a⋅4 a=2
ƒ(x) = 2 ⋅ 2 x
KEY EXAMPLE
Write the general form. Substitute the value for b. Substitute a pair of input-output values. Simplify. Solve for a. Use a and b to write an equation for the function.
(Lesson 14.5)
Describe the transformations of the function g(x) = 2(3) + 5 as compared to the parent function f(x) = 3 x. x
a has changed from 1 to 2. This corresponds to a vertical stretch by a factor of 2. The constant has changed from 0 to 5. This corresponds to a translation of 5 units up. g(x) has been stretched by a factor of 2 and translated 5 units up.
Module 14
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Study Guide Review
SCAFFOLDING SUPPORT
IN1_MNLESE389762_U6M14MC 703
• Elicit from students that fewer friends require more time and more friends require less time to meet the goal. • For students who need more structure, provide a range of 9 to 15 friends in 6 to 7 days. • This task could be made more challenging by suggesting that only a certain percentage of friends will share the video within the time frame. Ask how this changes the number of original contacts.
703
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EXERCISES Find the common ratio r for each geometric sequence and use r to find the next three terms. (Lesson 14.1) 1.
1701, 567, 189,…
SAMPLE SOLUTION Assumptions:
5, 20, 80,…
2.
1 ; 63, 21, 7 r = __ 3
• The video takes a full week to reach 750,000 views.
r = 4; 320, 1280, 5120
Write a recursive rule and an explicit rule for each geometric sequence. (Lesson 14.2) 3.
4, 12, 36, 108, 324,…
f(1) = 4 and f(n) = f(n - 1) ⋅ 3;
f(1) = 6 and f(n) = f(n - 1) ⋅ 5;
f(n) = 4 ⋅ 3
f(n) = 6 ⋅ 5 n - 1
n-1
First, find the relationship between friends, days, and number of views. If you show the video to n friends on Day 1 and they each show it to n people, then on Day 2, n · n = n 2 people will see it for the first time. On Day 3, n 2 · n = n 3 people will see it, and so on. So the relationship is:
6, 30, 150, 750, 3750,…
4.
Write an equation for the exponential function that includes the pair of given points. (Lesson 14.3) 5.
(2, 16) and (3, 32)
6.
f(x) = 4 ⋅ 2 x 7.
(2, 4) and (3, 2)
views = (friends)
days
1 f(x) = 16 ⋅ (__ )
x
2
Now use a guess and check strategy to see how many friends are needed for the video to hit the goal on Day 7. Make a table with the help of a calculator to find the least value of n that satisfies the inequality n 7 ≥ 750,000.
Find a, b, and the y-intercept for ƒ(x) = 5(2) , and then describe its end behavior. (Lesson 14.4) x
a = 5; b = 2; y-intercept = 5; end behavior: As x → ∞, y → ∞ and as x → -∞, y → 0. MODULE PERFORMANCE TASK
10 7
What Does It Take to Go Viral?
© Houghton Mifflin Harcourt Publishing Company
You want your newest video to be so popular that it gets more than 750,000 daily views within a week after you post it. You share it with friends and assume that each friend will share the video with the same number of people that you do, and so on. How can you determine the smallest number of friends you need to show your video to? What answer do you think would be too big? Too small? Start by listing in the space below how you plan to tackle the problem. Then use your own paper to complete the task. Be sure to write down all your data and assumptions. Then use numbers, tables, or algebra to explain how you reached your conclusion.
Module 14
704
10,000,000 Too high
9
7
4,782,969 Too high
8
7
2,097,152 Acceptable
7
7
823,543 Best
6
7
279,936 Too low
So, you would need to show the video to 7 friends on the first day.
Study Guide Review
DISCUSSION OPPORTUNITIES
IN1_MNLESE389762_U6M14MC 704
4/19/14 11:45 AM
• Which, if any, parameters can change without affecting the number of friends you first share with? • Have students compare and evaluate different strategies and assumptions that classmates used to solve the problem. Assessment Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem.
Study Guide Review 704
Ready to Go On?
Ready to Go On?
ASSESS MASTERY
14.1–14.5 Geometric Sequences and Exponential Functions
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
• Online Homework • Hints and Help • Extra Practice
Write a recursive rule and an explicit rule for each geometric sequence, and then find the next three terms. (Lessons 14.1, 14.2) 2, 8, 32,…
1.
ASSESSMENT AND INTERVENTION
2.
1024, 512, 256,…
1 f(1) = 1024 and f(n) = f(n - 1) ⋅ __ 2 n-1 1 f(n) = 1024 ⋅ __ 2 128, 64, 32
f(1) = 2 and f(n) = f(n - 1) ⋅ 4
()
f(n) = 2 ⋅ 4 n - 1 128, 512, 2048
Write an equation for the exponential function that includes the pair of given points. Find a, b, and the y-intercept, and then graph the function and describe its end behavior. (Lessons 14.3, 14.4)
(1, 12) and (–1, 0.75)
3.
f(x) = 3 ⋅ 4 x ; a = 3; b = 4; y-int = (0, 3);
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
As x → ∞, y → ∞ and as x → -∞, y → 0. 20
Differentiated Instruction Resources • Reading Strategies • Success for English Learners • Challenge Worksheets Assessment Resources
-2 0 -10
-4
20
y
10 2
x
4
-4
-2
0
2
4
-20
Describe the transformations of the function g(x) = 0.25(5) - 2 as compared to the parent function ƒ(x) = 5 x. (Lesson 14.5) x
5.
g(x) has been vertically compressed by a factor of 0.25 and translated 2 units down.
ESSENTIAL QUESTION How does the rate of change of an exponential function behave as the value of x increases?
6.
Possible Answer: As x increases, the rate of change of an exponential function is either consistently increasing or consistently decreasing.
Module 14
COMMON CORE IN1_MNLESE389762_U6M14MC 705
Module 14
and as x → -∞, y → -∞.
y
-20
• Leveled Module Quizzes
705
()
x © Houghton Mifflin Harcourt Publishing Company
• Reteach Worksheets
(–1, –8) and (1, –2) x 1 ; a = -4; b = __ 1; f(x) = -4 __ 2 2 y-int = (0, -4); As x → ∞, y → 0
10
ADDITIONAL RESOURCES Response to Intervention Resources
4.
Study Guide Review
705
Common Core Standards
4/19/14 11:45 AM
Content Standards Mathematical Practices
Lesson
Items
13.1, 13.2
1
F-LE.A.2, F-BF.A.1
MP.7
13.1, 13.2
2
F-LE.A.2, F-BF.A.1
MP.7
13.3, 13.4
3
F-LE.A.2, F-IF.A.2, F-IF.C.7
MP.7
13.3, 13.4
4
F-LE.A.2, F-IF.A.2, F-IF.C.7
MP.7
MODULE MODULE 14 MIXED REVIEW
14
MIXED REVIEW
Assessment Readiness
Assessment Readiness
1. Consider the geometric sequence 6, 24, 96, 384, …. Choose True or False for each statement. A. The sixth term is 1536. n-1 B. The explicit rule is f(n) = 6(4) . C. The recursive rule is f(1) = 4; f(n) = f(n-1) · 6.
True True
False False
True
False
ASSESSMENT AND INTERVENTION
2. Does the given system of equations have exactly one solution? Select Yes or No for each system. ⎧3x-2y = 6 A. ⎨ ⎩2x+ 2y = 14
Yes
No
⎧ y = -4x - 5
Yes
No
⎧5x - 3y = 15
Yes
No
B. ⎨
⎩ y = -4x + 2
C. ⎨
⎩ 5x + 3y = 15
Assign ready-made or customized practice tests to prepare students for high-stakes tests.
ADDITIONAL RESOURCES
3. Is the given number a term in both the sequence f (n) = f (n - 1) + 5 and the sequence n-1 f(n) = 3(2) , if f(1) = 3? Select Yes or No for each number. A. B. C. D.
8 18 24 48
Yes Yes Yes Yes
• Leveled Module Quizzes: Modified, B
p(n) = 6(0.98) ; a is the original output of the beam, and b is 1 - 0.02, or the percent of power retained after each reflection. n
COMMON CORE
AVOID COMMON ERRORS Item 2 Some students will attempt to solve each system the same way, even though each system is set up such that they are solved most easily using different methods. Remind students to consider each system individually when choosing a solution method.
© Houghton Mifflin Harcourt Publishing Company
4. A laser beam with an output of 6 milliwatts is focused at a series of mirrors. The laser beam loses 2% of its power every time it reflects off of a mirror. The power p(n) is an exponential function of the number of reflections in the form of p(n) = ab n. Write the equation p(n) for this laser beam. Explain how you determined the values of a and b.
Module 14
Assessment Resources
No No No No
Study Guide Review
706
Common Core Standards
IN1_MNLESE389762_U6M14MC 706
4/19/14 11:45 AM
Content Standards Mathematical Practices
Lesson
Items
13.1, 13.2
1
F-LE.A.2
MP.4
11.1
2*
A-REI.C.6
MP.1
4.1, 13.1
3*
F-LE.A.2
MP.1
13.3
4
F-IF.A.2
MP.4
* Item integrates mixed review concepts from previous modules or a previous course.
Study Guide Review 706
MODULE
15
15 MODULE
Exponential Equations and Models
Exponential Equations and Models
Essential Question: How can you use exponential
ESSENTIAL QUESTION:
equations to represent real-world situations?
Answer: Many real-world situations cannot be represented by linear functions. Exponential equations can be used to represent situations in which the output values increase or decrease by a constant ratio for each unit increase in the input value.
LESSON 15.1
Using Graphs and Properties to Solve Equations with Exponents LESSON 15.2
Modeling Exponential Growth and Decay LESSON 15.3
Using Exponential Regression Models LESSON 15.4
This version is for PROFESSIONAL DEVELOPMENT Algebra 1 and Geometry only VIDEO
Comparing Linear and Exponential Models
Author Juli Dixon models successful teaching practices in an actual high-school classroom.
Professional Development my.hrw.com
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Everett Collection Inc/Alamy
Professional Development Video
REAL WORLD VIDEO Scientists have found many ways to use radioactive elements that decay exponentially over time. Uranium-235 is used to power nuclear reactors, and scientists use Carbon-14 dating to calculate how long ago an organism lived.
MODULE PERFORMANCE TASK PREVIEW
Half-Life
Accidents at nuclear reactors like the one in Fukushima, Japan, in 2011 commonly release the radioactive isotopes iodine-131 and cesium-137. Iodine-131 often causes thyroid problems, whereas cesium-137 permeates the entire body and can cause death. Each isotope decays over time but at very different rates. How can you figure out the concentration of isotopes at a nuclear accident? Let’s find out!
Module 15
DIGITAL TEACHER EDITION IN1_MNLESE389762_U6M15MO 707
Access a full suite of teaching resources when and where you need them: • Access content online or offline • Customize lessons to share with your class • Communicate with your students in real-time • View student grades and data instantly to target your instruction where it is needed most
707
Module 15
707
PERSONAL MATH TRAINER Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, Common Core-aligned practice tests.
19/04/14 10:23 AM
Are YOU Ready?
Are You Ready?
Complete these exercises to review skills you will need for this chapter.
Constant Rate of Change
ASSESS READINESS
Tell if the rate of change is constant.
Example 1
+1
Time (hr)
1
Distance (mi) 45
+1 +1
2
3
4
90
135
180
Use the assessment on this page to determine if students need strategic or intensive intervention for the module’s prerequisite skills.
• Online Homework • Hints and Help • Extra Practice
change in miles rate of change = _______________ change in hours 45 = ___ 1
+45 +45 +45 The rate of change is constant.
ASSESSMENT AND INTERVENTION
Tell if the rate of change is constant. 1.
Age (mo)
Weight (lb)
3
6
9
12
12
16
18
20
2.
Hours
Pay ($)
2
4
6
8
16
32
48
64
yes
no
3
Percent
2 1
21% 0.21
4.
3.5% 0.035
5.
108% 1.08
6.
0.25% 0.0025
Exponents Find the value of 2(3) . 4
4 2(3) = 2(3 ⋅ 3 ⋅ 3 ⋅ 3)
Write the power as a multiplication expression. Multiply.
= 162
Find the value. 7.
3(4) 48
9.
5(2) 160
2
8.
Module 15
Tier 1 Lesson Intervention Worksheets Reteach 15.1 Reteach 15.2 Reteach 15.3 Reteach 15.4
ADDITIONAL RESOURCES See the table below for a full list of intervention resources available for this module. Response to Intervention Resources also includes: • Tier 2 Skill Pre-Tests for each Module • Tier 2 Skill Post-Tests for each skill
3
3 10. 350(0.1) 0.35
5
IN1_MNLESE389762_U6M15MO 708
24(0.5) 3
© Houghton Mifflin Harcourt Publishing Company
3.
Example 3
Personal Math Trainer will automatically create a standards-based, personalized intervention assignment for your students, targeting each student’s individual needs!
r and Write the percent in the form ____ 100 then write the decimal.
Write 7% as a decimal. 7 = 0.07 7% = _ 100 Write the percent as a decimal. Example 2
TIER 1, TIER 2, TIER 3 SKILLS
708
Response to Intervention Tier 2 Strategic Intervention Skills Intervention Worksheets 4 Constant Rate of Change 5 Exponents 15 Percent 18 Scatter Plots
Differentiated Instruction
19/04/14 10:23 AM
Tier 3 Intensive Intervention Worksheets available online Building Block Skills 5, 24, 27, 29, 30, 37, 39, 46, 59, 63, 65, 68, 69, 70, 72, 76, 100
Challenge worksheets Extend the Math Lesson Activities in TE
Module 15
708
LESSON
15.1
Name
Using Graphs and Properties to Solve Equations with Exponents
In previous lessons, variables have been raised to rational exponents and you have seen how to simplify and solve equations containing these expressions. How do you solve an equation with a rational number raised to a variable? In x certain cases, this is not a difficult task. If 2 x = 4 it is easy to see that x = 2 since 2 2 = 4. In other cases, like 3(2) = 96, where would you begin? Let’s find out.
Create equations ...and use them to solve problems… arising from… exponential functions. Also A-SSE.B.3c, A-REI.D.11, F-BF.A.1, F-LE.A.2
A B
Mathematical Practices
Solve for 3(2) = 96 for x. x
Let ƒ(x) = 3(2) . Complete the table for ƒ(x). x
MP.5 Using Tools
Language Objective
x
f(x)
1
6
2
Explain to a partner how to use a graph to find the solution to an equation with a variable exponent.
3 4 5 6
When the bases are equal, use the Equality of Bases Property. When the bases are not equal, graph each side of the equation as its own function and find the intersection.
PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss why a town government might need to know the rate at which the town’s population is growing. Then preview the Lesson Performance Task.
7 © Houghton Mifflin Harcourt Publishing Company
Essential Question: How can you solve equations involving variable exponents?
Solving Exponential Equations Graphically
Explore 1
A-CED.A.1
ENGAGE
D
g(x)
1
96
2
6
96 96 96 96 96
7
96
4 5
F
12 24 48 96 192 384
x
The graphs intersect at point(s):
Module 15
72 48
(4, 48) f(x)
24 (3, 24) (1, 6) (2, 12) 0
-2
x
1 2 3 4 5 6 7 8
Using the table, graph g(x) on the same axes as ƒ(x). y
E
96 72
g(x)
(5 , 96) f(x)
48 24 x 0
-2
1 2 3 4 5 6 7 8
(5, 96) This means that ƒ(x) = g(x) when x = 5 . be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
Lesson 1
709
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made throu
Date Class
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Name
IN1_MNLESE389762_U6M15L1 709
Resource Locker
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96 96 96 96 96
(5 , 96) f(x)
48 x
24 -2
0
96
at point( s intersect The graph
Module 15
g(x)
72
96
7
IN1_MNL
g(x table, graph Using the as ƒ(x). same axes y
g(x)
x
(3, 24) x 24 (2, 12) (1, 6) 8 5 6 7 0 1 2 3 4 ) on the
lete the table 96. Comp
Let g(x) = for g(x).
(4, 48) f(x)
48
-2
Watch for the hardcover student edition page numbers for this lesson.
72
6
12 24 48 96 192 384
2
HARDCOVER PAGES 709720
table of values Using the provided. on the axes y (5, 96) 96
f(x)
1
Lesson 15.1
(5, 96)
96
Let g(x) = 96. Complete the table for g(x).
3
Using the table of values, graph ƒ(x) on the axes provided. y
C
x
709
Resource Locker
Essential Question: How can you solve equations involving variable exponents?
The student is expected to:
COMMON CORE
Date
15.1 Using Graphs and Properties to Solve Equations with Exponents
Common Core Math Standards COMMON CORE
Class
s):
(5, 96)
This means
that ƒ(x)
7 8 4 5 6 1 2 3
when x = = g(x)
5 . Lesson 1
709 19/04/14
10:37 AM
19/04/14 10:41 AM
Reflect
EXPLORE 1
Discussion Consider the function h(x)= -96. Where do ƒ(x) and h(x) intersect? The graphs would not intersect as f(x) is always greater than 0. Raising any positive
1.
Solving Exponential Equations Graphically
number to a positive exponent yields a positive number. Divide the equation by 3( 2 = 96 by 3 on both sides (an Algebraic Step) and utilize the same method as )x
2.
above. The point of intersection would be:
(5, 32)
.
Is this the same point of intersection? Is this the same answer? Can this be done? Elaborate as to why or why not.
INTEGRATE TECHNOLOGY
It is not the same point of intersection. The y-values of the points are different. They
Students can use graphing calculators to solve an exponential equation by the method shown in the Explore activity. Students should enter the appropriate exponential function and constant function, graph both functions, and use the calculator’s intersect feature to find their point of intersection.
do represent the same solution because the equations are equivalent by the Division Property of Equality.
Solving Exponential Equations Algebraically
Explore 2
Recall the example 2 = 4, with the solution x = 2. What about a slightly more complicated equation? Can an x equation like 5 (2) = 160 be solved using algebra? x
A
Solve 5 (2) = 160 for x. The first step in isolating the term containing the variable x
on one side of the equation is to divide each side of the equation by 5 .
QUESTIONING STRATEGIES
5( 2 ) x _ _ = 160
5
B
When solving an exponential equation of the form ab x = c graphically, what two functions do you graph? the exponential function f(x) = ab x and the horizontal line g(x) = c
5
Simplify.
C
(2) = 32 x
Rewrite the right hand side as a power of 2. 5 x (2) = (2)
D
Solve. x= 5
3.
Discussion The last step of the solution process seems to imply that if b x = b y then x = y. Is this true for all values of b? Justify your answer. No, it is not true. For example, 0 5 = 0 8 but 5 ≠ 8, or 1 7 = 1 958 but 7 ≠ 958.
4.
In Reflect 2, we started to solve 3(2) = 96 algebraically. Finish solving for x. x 3(2) = 96 x
2 x = 32 2x = 25
How does graphing these two functions on the same grid help you determine the value of the exponent? The value of the exponent is the x-value of the point of intersection.
© Houghton Mifflin Harcourt Publishing Company
Reflect
EXPLORE 2 Solving Exponential Equations Algebraically
x=5
QUESTIONING STRATEGIES Module 15
710
Lesson 1
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U6M15L1 710
19/04/14 10:41 AM
Learning Progressions In this lesson, students continue to build on their understanding of geometric sequences and exponential functions. They learn the Equality of Bases Property, which states that If b > 0 and b ≠ 1, then b x = b y if and only if x = y. They learn to solve equations involving variable exponents either by using the Equality of Bases Property or by graphing. They also begin to model real-world situations using exponential equations, which can then be solved by either method. Work with exponential functions will continue as students learn about exponential growth and decay models and exponential regression.
In the equation 4 x = 64, how can you evaluate x? Write 64 as a power of 4. 3 64 = 4 , so x = 3. Assuming that x is an integer in the equation b x = c, what must be true for this method to work? The value of c must be a power of b.
Using Graphs and Properties to Solve Equations with Exponents
710
Solving Equations by Equating Exponents
Explain 1
EXPLAIN 1
Solving the previous exponential equation for x used the idea that if 2 x = 2 5, then x = 5. This will be a powerful tool for solving exponential equations if it can be generalized to if b x = b y then x = y. However, there are values for which this is clearly not true. For example, 0 7 = 0 3 but 7 ≠ 3. If the values of b are restricted, we get the following property.
Solving Equations by Equating Exponents
Equality of Bases Property Two powers with the same positive base other than 1 are equal if and only if the exponents are equal. Algebraically, if b > 0 and, b ≠ 1, then b x = b y if and only if x = y.
QUESTIONING STRATEGIES x
5x = 54
5· Multiply both sides by _ 2 Simplify. Rewrite the right side as a power of 5.
x = 4
Equality of Bases Property.
( )
5 x=_ 250 2 _ 27 3
( )
5 x 250 _ 2 _ 3 27 _ =_
2
Divide both sides by 2 .
2
(_53 ) © Houghton Mifflin Harcourt Publishing Company
Equality of Bases Property. Have students give examples to show why the property does not apply when the base is 0, 1, or –1. For example: 05 = 0 8 = 0 4 but 5 ≠ 8; 1 20 = 1 99 = 1 but 20 ≠ 99; and (–1) 6 = (–1) = 1 but 4 ≠ 6.
2 ( 5 ) x = 250 _ 5 5·_ 5 2 (5) x = 250 · _ _ 2 5 2 5 x = 625
How does this step compare to isolating a variable on one side of a linear equation? It is done for the same reason. By isolating the power, you have isolated the variable as well. Then you can compare the exponents in the final equivalent expression.
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Discuss with students the limitations on the
Solve by equating exponents and using the Equality of Bases Property.
Example 1
In an equation such as 36(2) = 576, what property of equality can you use to isolate 2 x? Explain how. Division Property of Equality; divide both sides by 36.
(_53 )
x
x
125 =_ 27 =
(_53)
Simplify.
3
x= 3
Rewrite the right side as a power of
5 _ . 3
Equality of Bases Property.
Reflect
5.
Suppose while solving an equation algebraically you are confronted with: 5 x = 15 5x = 5 Can you find x using the method in the examples above?
No, you cannot. It is not possible because 15 is not a whole number power of 5.
AVOID COMMON ERRORS Some students may misread the base b in an expression b x as a coefficient of x and try to divide both sides of the equation by b to isolate the variable. Remind them that when a number is raised to a power, it cannot be treated as a single factor. They must use the properties of equality to isolate b x, then use the Equality of Bases Property to solve for the variable.
711
Lesson 15.1
Module 15
711
Lesson 1
COLLABORATIVE LEARNING IN1_MNLESE389762_U6M15L1 711
Peer-to-Peer Activity
Have students work in pairs. Have each student write an equation involving a variable exponent in the form b x = c. After students exchange equations, each partner should first decide whether c can be expressed as a whole number power of b. If so, the student should rewrite c as a power of b and solve for x. If not, the student should use a graphing calculator to graph each side of the equation as a separate function and use the intersect feature to find the x-coordinate of the intersection point, which is the solution to the original equation. Have students check each other’s work.
19/04/14 10:41 AM
Your Turn
EXPLAIN 2
Solve by equating exponents and using the Equality of Bases Property. 6.
2 (3) x = 18 _ 3
7.
x
x
x
x
2
x x
3x = 33
QUESTIONING STRATEGIES
2
3 x= 2
x=3
Explain 2
Solving a Real-World Exponential Equation by Graphing
_3 (_4 ) = _8 _22 ⋅ 3_3 (_4 ) 3= _2 ∙ _8 3 2 3 3 3 16 (_434 ) = _ 9 (_) = (_4 )
_2 (3) = 18 _33 ⋅ _2 (3) = 18 ∙ _3 2 3 3 x = 27
()
x
8 3 _ 4 =_ _ 2 3 3
3
If a population grows by 5% each year, by what factor is the population multiplied each year? Explain. 1.05; if the population is p one year, it will be p + 0.05p = 1.05p the next year.
Solving a Real-World Exponential Equation by Graphing
Why is it appropriate to round a prediction involving time to the nearest year? A prediction is usually just an estimate, so rounding is appropriate.
Some equations cannot be solved using the method in the previous example because it isn’t possible to write both sides of the equation as a whole number power of the same base. Instead, you can consider the expressions on either side of the equation as the rules for two different functions. You can then solve the original equation in one variable by graphing the two functions. The solution is the input value for the point where the two graphs intersect.
Analyze Information Identify the important information. • The starting population is • The ending population is • The growth rate is
20,000. 40,000
.
8% or 0.08 .
Formulate a Plan With the given situation and data there is enough information to write and solve an exponential model of the population as a function of time. Write the exponential equation and then solve it using a graphing calculator. Set ƒ(x) = the target population and g(x) = the exponential model . Input Y 1 = ƒ(x) and Y 2 = g(x) into a graphing calculator, graph the functions, and find their intersection .
Module 15
712
© Houghton Mifflin Harcourt Publishing Company · Image Credits: © James Prout/Alamy
An animal reserve has 20,000 elk. The population is increasing at a rate of 8% per year. There is concern that food will be scarce when the population has doubled. How long will it take for the population to reach 40,000?
Lesson 1
DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U6M15L1 712
19/04/14 10:41 AM
Graphic Organizers Have students complete a graphic organizer that shows when to solve an equation involving a variable exponent algebraically and when to solve it graphically. Solving b x = c (where b > 0, b ≠ 1, and c > 0) Value of c Solution Method d c = b for some Algebraic: whole number d. c is not a power of b.
b x = b d, so x = d.
Graph: intersection of ƒ(x) = b x and g(x) = c
Using Graphs and Properties to Solve Equations with Exponents
712
Solve
INTEGRATE TECHNOLOGY
Write a function P(t) = ab t, where P is the population and t is the number of years since the population was initially measured.
When solving exponential equations graphically, have a student demonstrate how to identify the two functions to be graphed, enter them into a graphing calculator, and find the solution by finding the point of intersection. Discuss how to adjust the viewing window so that the graph and the point of intersection are clearly visible.
a represents the initial population of elk a = 20,000 b represents the yearly growth rate of the elk population b = 1.08
( 1.08 ) . t
The function is P(t) =
20,000
() To find the time when the population is 40,000, set the function or P t equal to 40,000 and solve for t .
( 1.08 ) . t
40,000 = 20,000
Write functions for the expressions on either side of the equation. 40,000
ƒ(x) =
) ( g (x) = 20,000 1.08
x
Using a graphing calculator, set Y 1 = ƒ(x) and Y 2 = g (x) View the graph. Use the intersect feature on the CALC menu to find the intersection of the two graphs. The approximate x-value where the graphs intersect is 9.006468 . Therefore, the population will double in just a little over 9 years.
Justify and Evaluate © Houghton Mifflin Harcourt Publishing Company
Check the solution by evaluating the function at t = 9 . P
( 9 ) = 20,000 ⋅ (1.08) = 20,000 ⋅ ( 1.9990 ) 9
= 39,980 Since 39,980 ≈ 40,000, it is in 9 years.
accurate to say the population will double
This prediction is reasonable because 1.08
Module 15
9
713
≈ 2 .
Lesson 1
LANGUAGE SUPPORT IN1_MNLESE389762_U6M15L1 713
Connect Context
19/04/14 10:41 AM
Support students in interpreting the language used in problem statements. Explain that the word suppose at the beginning of a problem signals that what follows is a hypothetical example, meaning that readers should use their imaginations to consider a possible scenario. Often, a problem will be followed by the question, Why or why not? Explain that the question is phrased this way so as not to give away the answer. Students should understand that they need to explain either why a result is true or why it is not true, depending on the situation.
713
Lesson 15.1
Your Turn
ELABORATE
Solve using a graphing calculator. 8.
9.
There are 225 wolves in a state park. The population is increasing at the rate of 15% per year. You want to make a prediction for how long it will take the population to reach 500. Y 1 = 500 Graph The intersection point is (5.713341, 500). The wolf x Y 2 = 225(1.15) population will reach 500 in approximately 5.7 years. There are 175 deer in a state park. The population is increasing at the rate of 12% per year. You want to make a prediction for how long it will take the population to reach 300. Y 1 = 300 The intersection point is (4.756046, 300). The deer Graph x Y 2 = 175(1.12) population will reach 300 in approximately 4.8 years.
QUESTIONING STRATEGIES What is the shape of the graph of an exponential function of the form f(x) = b x when b > 1? It is a curve that rises in greater and greater amounts as x increases. What is the shape of the graph of a function ƒ(x)= b x when 0 < b < 1? It is a curve that falls more and more gradually as x increases.
Elaborate 10. Explain how you would solve 0.25 = 0.5 x. Which method can always be used to solve an exponential equation? Possible answer: Algebraically.
0.25 = 0.5 x
(0.5)2 = (0.5) → x = 2
SUMMARIZE THE LESSON
Exponential equations can always be solved graphically.
How can you solve an equation where the variable is an exponent? First, use the properties of equality to isolate the number raised to a variable power. Then check whether the constant on the other side of the equation can be written as a power of the same base. If it can, use the Equality of Bases Property to solve. If not, graph each side of the equation as its own function and find the x-value of the point of intersection.
x
x 11. What would you do first to solve the equation __14 (6) = 54?
Multiply each side of the equation by 4 to isolate the power.
12. How does isolating the power in an exponential equation like __14 (6) = 54 compare to isolating the variable in a linear equation? Both are done for the same reason. By isolating the power, you have isolated the variable x
as well. Then you can compare the exponents in the final equivalent expressions.
situation, 2 = 0.99 x, has no solution for x > 0. The graphing calculator will show a horizontal line at 2 and an exponential function with a y-intercept of 1 decreasing towards the positive x-axis. 14. Solve 0.5 = 1.01 x graphically. Suppose this equation models the point where a population increasing at a rate of 1% per year is halved. When will the population be halved? Since x = -69.66072, you would have to go back in time, which is not possible. Seventy
years or so ago the population was half of what it is now.
© Houghton Mifflin Harcourt Publishing Company
13. Given a population decreasing by 1% per year, when will the population double? What will this type of situation look like when graphed on a calculator? It will never double as the population is decreasing. The equation representing this
15. Essential Question Check-In How can you solve equations involving variable exponents? When the bases are equal, use the Equality of Bases Property. When there are not equal
bases on both sides of the equation, graph each side of the equation as its own function and find the intersection. Module 15
IN1_MNLESE389762_U6M15L1 714
714
Lesson 1
19/04/14 10:41 AM
Using Graphs and Properties to Solve Equations with Exponents
714
Evaluate: Homework and Practice
EVALUATE 1.
• Online Homework • Hints and Help • Extra Practice
Would it have been easier to find the solution to the equation in Explore 1, x 3(2) = 96, algebraically? Justify your answer. In general, if you can solve an exponential equation graphing by hand, why can you solve it algebraically?
Yes, 3(2) = 96 becomes (2) = 32 after dividing both sides of the equation by 3 and 32 is an integer power of 2. x
x
In general, the input-output tables for f(x) and g(x) have integers in the domain and the values in the range are easy to calculate.
ASSIGNMENT GUIDE Concepts and Skills
Practice
Explore 1 Solving Exponential Equations Graphically
Exercises 2–3
Explore 2 Solving Exponential Equations Algebraically
Exercises 1, 3
Example 1 Solving Equations by Equating Exponents
Exercises 4–16, 24 –25
Example 2 Solving a Real-World Exponential Equation by Graphing
Exercises 17–23
2.
The equation 2 = (1.01) models a population that has doubled. What is the rate of increase? What does x represent? x
The rate of increase is 1% per unit time. x is number of units of time. 3.
Can we solve equations using both algebraic and graphical methods?
Yes. We can simplify the equation algebraically and then use graphing. Solve the given equation. x 4. 4(2) = 64
5.
4(2) 64 ____ = __ 4
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Circulate as students solve the practice exercises. Invite students to explain their reasoning as they begin a new problem.
© Houghton Mifflin Harcourt Publishing Company
7.
7(3) = 63 x
7
3x = 9
2x = 24
3x = 32
x=4
x=2
75 (_14 )(_56 ) = _ 432 x
8.
5 75 1 _ = 4 ⋅ ___ 4⋅ _ 4 6 432 x
x
x
2
x=2
()
Exercise
715
Lesson 15.1
6 x = 216 6x = 63 x=3
x
49 7 =_ 2_ 2 2
9.
2( 2 ) 2 ____ = __
_7
2 _7 2 _7 2
x
2 49 __ = 4 x 7 = _ 2
3(11) = 3993 x
3(11) 3993 ______ = ____ x
49 __
3
3
() () ()
11 x = 1331
x
11 x = 11 3
2
x=3
x=2
Module 15
IN1_MNLESE389762_U6M15L1 715
x
7
2 x = 16
6 x = 54 _ 4
6 = 4 · 54 4 · __ 4
x
4
( )( ) 25 (_56) = __ 36 5 (_6) = (_56)
6.
7(3) 63 ____ = __
x
Lesson 1
715
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1
2 Skills/Concepts
MP.6 Precision
2
2 Skills/Concepts
MP.4 Modeling
3
2 Skills/Concepts
MP.6 Precision
4–12
1 Recall of Information
MP.2 Reasoning
13–16
2 Skills/Concepts
MP.2 Reasoning
17–22
2 Skills/Concepts
MP.4 Modeling
19/04/14 10:41 AM
()
x
1 2 _ 9
9x = 92
2
9
81
(_21 )(_32 ) = (_41 )(_1627 ) x
( )( ) ( )( ) 8 (_23) = __ 27 (_23) = (_23)
16 1 _ 2 1 __ 2 _ =2 _ 4 27 2 3 x
x
x
3
x=3
16.
(_25) (_25) (_25)
2x
2x
2x
x
( )
15.
x
16 ___
8
8
3
27
13
variable exponent, suggest that students try to structure the solution so that they are solving an equation of the form b x = c y. If c = b, then x = y; if c ≠ b, then they should solve by graphing.
8 (_25 )(_25 ) = _ 125 x
x
x
x (_2) = (__8 )
x
x 3 (_2) = (_2)
3
169
8 (_52)(_25)(_25) = (_52)___ 125 4 (_25) = __ 25 (_25) = (_25)
8( 3 ) 4( 27 ) ____ = ____
_2
13
x=2
16 2 = (4) _ 14. (8) _ 27 3
8 _ 8 (_25 ) (_25 ) = (_ 125 )( 125 ) x
()
2
13
9
x=2
2
x 2 (__4 ) = (__4 )
x 2 (_1) = (_1)
9
x
x 16 (__4 ) = ___
x (_1) = __1
x=2
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 When solving an equation involving a
32 ___ ( ) ___ ____ = 169 4 2 __ 13
x
2
x
32 4 =_ 12. 2 _ 13 169
2 __ ( ) __ ____ = 81
9 x = 81
13.
( )
x
1 =_ 2 11. 2 _ 81 9
10. 2(9) = 162
3
2
x=2
x=3
x
( ) 2 = ((_ 5) ) 2 _ = (5)
8 = ___ 125
2
3 2
6
© Houghton Mifflin Harcourt Publishing Company · Image Credits: ©prudkov/ Shutterstock
2x = 6 6 2x __ =_ 2
2
x=3 17. There is a draught and the oak tree population is decreasing at the rate of 7% per year. If the population continues to decrease at the same rate, how long will it take for the population to be half of what it is? The model for the oak tree population
is P(t) = P i(0.93) , where t is the time in years, P i is the initial population, and P is the population in year t. To find when the population is half of its initial value, solve Pi for t. P(t) = __ 2 t
P t __ = P i(0.93) i
2
P __
P (0.93) __2 = ______ i
t
i
Pi Pi _1 = (0.93) t 2
t ≈ 9.55
The population will reach half of its original value in approximately 9.6 years.
Module 15
Exercise
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Lesson 1
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Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
23
3 Strategic Thinking
MP.4 Modeling
24–25
3 Strategic Thinking
MP.3 Logic
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Using Graphs and Properties to Solve Equations with Exponents
716
18. An animal reserve has 40,000 elk. The population is increasing at a rate of 11% per year. How long will it take for the population to reach 80,000?
AVOID COMMON ERRORS
The model for population is P(t) = 40,000(1.11) , where t is the time in t
Students may be confused by complicated equations that involve variable exponents as well as additional factors. Remind them to first apply the properties of equality to isolate the number with the variable exponent, then use the Equality of Bases Property to solve.
years and P is the population in year t. To find when the population is 80,000, solve P(t) = 80,000 for t. 80,000 = 40,000(1.11)
t
40,000(1.11) 80,000 _____ = _________ t
40,000
40,000
2 = (1.11)
t
t ≈ 6.64 The population will reach 80,000 in approximately 6.6 years. 19. A lake has a small population of a rare endangered fish. The lake currently has a population of 10 fish. The number of fish is increasing at a rate of 4% per year. When will the population double? How long will it take the population to be 80 fish? t The model for population is P(t) = 10(1.04) , where t is the time in years
and P is the population in year t. Solve P(t) = 20 for t.
20 = 10(1.04)
t
10(1.04) 20 __ = ______ t
10
10
2 = 1.04 t t ≈ 17.67 The population of the fish will double in 18 years. To find when the population will be 80, you can solve P(t) = 80 for t.
© Houghton Mifflin Harcourt Publishing Company
Alternatively, note that 80 = 10 · 8 = 10 · 2 3. This corresponds to the population doubling three times, from 10 to 20, from 20 to 40, and from 40 to 80. The population will be 80 in 54 years (3 · 18). 20. Tim has a savings account with the bank. The bank pays him 1% per year. He has $5000 and wonders when it will reach $5200. When will his savings reach $5200?
The model is S(t) = 5000(1.01) . Solve S(t) = 5200 for t. t
5200 = 5000(1.01)
t
5000(1.01) 5200 ____ = ________ t
5000
5000
1.04 = 1.01 t t ≈ 3.94
Graphing f(x) and g(x), we get the point of intersection (3.941648, 1.04). Rounding up and considering interest is calculated yearly, it will take Tim 4 years. Module 15
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Lesson 1
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21. Tim is considering a different savings account that pays 1%, but this time it is compounded monthly.
AVOID COMMON ERRORS Some students may be unsure how to raise a fraction to a power. Remind them that both the numerator and the denominator must be raised to the same power.
(When interest is compounded monthly, the bank pays interest every month instead nt of every year. The function representing compounded interest is S(t) = P(1 + __nr ) , where P is the principal, or initial deposit in the account, r is the interest rate, n is the number of times the interest is compounded per year, t is the year, and S(t) is the savings after t years.) How many years will it take Tim to earn $200 at this bank? Should he switch?
(
)
0.01 The model is S(t) = 5000 1 + ___ 12
5200 = 5000(1.00083)
12t
12t
12t or S(t) = 5000(1.00083) . Solve S(t) = 5200 for t.
CURRICULUM INTEGRATION
5000(1.00083) 5200 ____ = ___________ 12t
5000
Encourage students to research applications of exponential functions. They should consider applications in science and business as well as uses in other math courses.
5000
1.04 = (1.00083)
12t
12t = 47.08 t = 3.92 Graphing f(x) and g(x), we get the point of intersection (3.923705, 1.04). x is approximately 4 years. Both accounts will reach $5200 an about 4 years.
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 As students solve real-world problems
Switching won’t make much difference. 22. Lisa has a credit card that charges 3% interest on a monthly balance. She buys a $200 bike and plans to pay for it by making monthly payments of $100. How many months will it take her to pay it off? Assume the first payment she makes is charged no interest because she paid it before the first bill. Her first payment is $100. At that time she owes $100 plus interest or $103. The second month she pays $100 and the third month she pays the rest. It takes her three months to pay it off. You do not have to solve an exponential because 3% is not a very high interest.
© Houghton Mifflin Harcourt Publishing Company
23. Analyze Relationships A city has 175,000 residents. The population is increasing at the rate of 10% per year. a. You want to make a prediction for how long it will take for the population to reach 300,000. Round your answer to the nearest tenth of a year. b. Suppose there are 350,000 residents of another city. The population of this city is decreasing at a rate of 3% per year. Which city’s population will reach 300,000 sooner? Explain.
a. 300,000 = 175,000(1.1)
x
x ≈ 5.7 The population will reach 300,000 in approximately 5.7 years. b. 300,000 = 350,000(.97)
involving time, have them make predictions before calculating their results. Write the predictions on the board, then compare them to the solutions found algebraically or graphically. Encourage students to improve their predictions by analyzing whether their predictions tend to be too high or too low and by considering how they can change their estimation methods.
x
x ≈ 5.1 5.1 < 5.7 The second city’s population will reach 300,000 sooner. Module 15
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Using Graphs and Properties to Solve Equations with Exponents
718
VISUAL CUES
H.O.T. Focus on Higher Order Thinking
24. Explain the Error Jean and Marco each solved the equation 9(3) = 729. Whose solution is incorrect? Explain your reasoning. How could the person who is incorrect fix the work? x
Have students create posters as visual reminders of how to solve equations involving exponents. Remind students to include examples as well as step-by-step procedures.
Jean
Marco
9(3) = 729
9(3) = 729
x
x
(_19 ) ⋅ 9(3) = (_19 ) ⋅ 729
3 2 ⋅ (3) = 729
x
x
3 x = 81 = 3 4
JOURNAL
3 2 + x = 729 = 3 6
x=4
In their journals, have students explain how to use the Equality of Bases Property to solve an equation with a variable exponent.
x=6
Jean is completely correct and Marco could correct his work as follows: Marco 9(3) = 729 x
3 2 ⋅ (3) = 729 x
2+x
3 = 729 = 3 6 x+2=6 x=4 He substituted for x = 6 instead of x + 2 = 6, which yields x = 4. 25. Critical Thinking Without solving, state the column containing the equation with the greater solution for each pair of equations. Explain your reasoning. 1 (3) x = 243 1 (9) x = 243 _ _ 3 3
()
()
1( ) The equation _ 3 = 243 has a greater solution. Since the values of the 3 x
© Houghton Mifflin Harcourt Publishing Company
powers of 3 increase less quickly than the values of the powers of 9, the 1( ) 1( ) value of x in _ 3 = 243 will be greater than the value of x in _ 9 = 243. 3 3 x
Module 15
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Lesson 15.1
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Lesson 1
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Lesson Performance Task
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Before students write an equation for the
A town has a population of 78,918 residents. The town council is offering a prize for the best prediction of how long it will take the population to reach 100,000. The population rate is increasing 6% per year. Find the best prediction in order to win the prize. Write an exponential equation in the form y = abx and explain what a and b represent.
situation in the Lesson Performance Task, discuss how they know that the base to be raised to a power in the exponential equation is 1.06 and not 0.06. Have them consider “What factor multiplied by the population makes the number 6% greater?” Then ask what the base would be if the population were decreasing by 6% per year. Students should recognize that it would be 1 – 0.06 = 0.94. Discuss what a graph showing each growth rate would look like.
Write an exponential equation. Let y represent the population and x represent time in years. a represents the initial population 78,918 b represents the rate of increase in the population per year y = 78,918(1 + 0.06)
x
Write the equations known as functions. f(x) = 100,000
g(x) = 78,918(1 + 0.06)
x
Use the intersect feature on a graphing calculator to find the point of intersection. The point of intersection is (4.063245, 100,000).
INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 As students use their graphing
The population will reach 100,000 in just over 4 years.
© Houghton Mifflin Harcourt Publishing Company · Image Credits ©Jim West/ Alamy Images
Module 15
720
calculators to graph the two functions and find their intersection, remind them to adjust the viewing window so that the intersection is shown clearly.
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have students share their reasons for why the point where the graphs of the right- and left-hand sides of the equation intersect is the solution.
Lesson 1
EXTENSION ACTIVITY IN1_MNLESE389762_U6M15L1 720
Have students research the current population of their community or state and the rate at which it is growing or decreasing. Then have students write an exponential equation in which y represents the population and x represents time in years. Finally, have students choose a future population size and predict when the population will reach that size.
19/04/14 10:40 AM
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Using Graphs and Properties to Solve Equations with Exponents
720
LESSON
15.2
Name
Modeling Exponential Growth and Decay
Class
Date
15.2 Modeling Exponential Growth and Decay Essential Question: How can you use exponential functions to model the increase or decrease of a quantity over time?
Common Core Math Standards
Resource Locker
The student is expected to: COMMON CORE
Explore
F-IF.C.7e
When you graph a function ƒ(x) in a coordinate plane, the x-axis represents the independent variable and the y-axis represents the dependent variable. Therefore, the graph of ƒ(x) is the same as the graph of the equation y = ƒ(x). You will use this form when you use a calculator to graph functions.
Graph exponential… functions, showing intercepts and end behavior… Also F-IF.B.5, F-BF.A.1a, F-LE.A.1c, F-LE.A.2
Mathematical Practices COMMON CORE
Use a graphing calculator to graph the exponential growth x function ƒ(x) = 200(1.10) , using Y 1 for ƒ(x). Use a viewing window from -20 to 20 for x, with a scale of 2, and from -100 to 1000 for y, with a scale of 50. Make a copy of the curve shown.
To describe the end behavior of the function, you describe the function values as x increases or decreases without bound. Using the TRACE feature, move the cursor to the right along the curve. Describe the end behavior as x increases without bound.
MP.4 Modeling
Apply mathematics to problems arising in everyday life, society, and Language Objective Compare and contrast exponential growth and exponential decay functions.
As x increases without bound, the graph grows larger at an exponentially increasing rate.
ENGAGE
An exponential function represents growth when b > 1 and decay when 0 < b < 1. You can use t y = a(1 + r) where a > 0 to represent exponential t growth and y = a(1 - r) where a > 0 to represent exponential decay.
PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss what an archeologist might learn from studying the bones, tools, weapons, and other objects left by prehistoric people. Then preview the Lesson Performance Task.
Using the TRACE feature, move the cursor to the left along the curve. Describe the end behavior as x decreases without bound.
As x decreases without bound, the graph approaches, but never hits, 0. © Houghton Mifflin Harcourt Publishing Company
Essential Question: How can you use exponential functions to model the increase or decrease of a quantity over time?
Describing End Behavior of a Growth Function
Reflect
1.
Describe the domain and range of the function using inequalities. Domain: {x| - ∞ < x < ∞} Range: {y| y < 0}
2.
Identify the y-intercept of the graph of the function. The y-intercept of the graph of the function is (0, 200).
3.
An asymptote of a graph is a line the graph approaches more and more closely. Identify an asymptote of this graph. The line y = 0 is an asymptote of this graph.
4.
Discussion Why is the value of the function always greater than 0? Since a positive number is being multiplied by another positive number, it is impossible for the growth function to go below the x-axis.
Module 15
be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
Lesson 2
721
gh “File info”
made throu
Date Class
owth nential Gr
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Name
se or Resource l the increa Locker ons to mode ential functi , use expon Also F-IF.B.5 can you behavior… ity over time? ion: How pts and end of a quant decrease showing interce functions, Function
Quest Essential
exponential… F-IF.C.7e Graph 1c, F-LE.A.2 F-LE.A. F-BF.A.1a,
IN1_MNLESE389762_U6M15L2 721
HARDCOVER PAGES 721734
a Growth
le and avior of ndent variab the ents the indepe graph of x-axis repres is the same as the plane, the Explore of ƒ(x) ns. a coordinate ore, the graph to graph functio n ƒ(x) in le. Theref tor graph a functio dent variab use a calcula When you the depen when you represents this form the y-axis You will use h y = ƒ(x). on ential growt equati the expon g
COMMON CORE
Describing
End Beh
Watch for the hardcover student edition page numbers for this lesson.
graph Use a viewin ator to x ing calcul Y 1 for ƒ(x). Use a graph shown. (1.10) , using a scale of 2, and of the curve ƒ(x) = 200 for x, with function Make a copy -20 to 20 scale of 50. as window from y, with a on values to 1000 for the be the functi from -100 cursor to on, you descri E feature, move the . of the functi TRAC ut bound behavior rate. . Using the ses witho be the end increasing without bound behavior as x increa To descri ses nentially or decrea the end r at an expo x increases Describe grows large the curve. the graph right along the end ut bound, Describe ases witho the curve. As x incre the left along to cursor e, move the hits, 0. . TRACE featur , but never without bound Using the aches ses appro as x decrea the graph behavior ut bound, ases witho As x decre
inequalities. on using of the functi y < 0} n and range Range: {y| the domai Describe < x < ∞} {x| - ∞ on. Domain: of the functi 200). tote of of the graph ion is (0, fy an asymp y-intercept of the funct closely. Identi Identify the the graph of and more rcept aches more The y-inte graph appro is a line the tote of a graph . 3. An asymp . of this graph ssible this graph asymptote 0? it is impo y = 0 is an greater than number, The line on always positive another of the functi plied by is the value multi Why ssion er is being 4. Discu . ive numb the x-axis Since a posit to go below th function Lesson 2 for the grow
y g Compan
Reflect
2.
© Houghto
n Mifflin
Harcour t
Publishin
1.
721 Module 15
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Explore 2
Describing End Behavior of a Decay Function
EXPLORE 1
Use the form from the first Explore exercise to graph another function on your calculator.
A
Use a graphing calculator to graph the exponential decay function x ƒ(x) = 500(0.8) , using Y 1 for ƒ(x). Use a viewing window from -10 to 10 for x, with a scale of 1, and from -500 to 5000 for y, with a scale of 500. Make a copy of the curve.
Describing End Behavior of a Growth Function
B
Using the TRACE feature, move the cursor to the right along the curve. Describe the end behavior as x increases without bound.
INTEGRATE TECHNOLOGY Students will use graphing calculators to complete the Explore activities that investigate the end behavior of exponential functions.
As x increases without bound, the graph approaches, but never hits, 0.
C
Using the TRACE feature, move the cursor to the left along the curve. Describe the end behavior as x decreases without bound.
As x decreases without bound, the graph grows larger at an exponentially increasing rate.
CONNECT VOCABULARY Connect the word bound in the phrase “as x increases without bound” to the more familiar word boundary. To increase without bound means to increase without reaching an end point or boundary. Another way to state the same idea is “as x approaches infinity.”
Reflect
5.
Discussion Describe the domain and range of the function using inequalities. Domain: {x| - ∞ < x < ∞} Range: {y| y < 0}
6.
7.
Identify the y-intercept of the graph of the function. The y-intercept of the graph of the function is (0, 500).
QUESTIONING STRATEGIES
line approaches, but never hits, y = 0.
Describe the x-intercepts of the exponential growth function. There are no x-intercepts for this function because there are no values of x for which f(x) = 0.
© Houghton Mifflin Harcourt Publishing Company
Identify an asymptote of this graph. Why is this line an asymptote? An asymptote of this graph is the line y = 0. It is an asymptote of the graph because the
Are there any asymptotes for this function other than the line y = 0? Explain. No; the graph gets closer and closer to y = 0 as x decreases without bound, and it is ever-increasing as x increases without bound.
EXPLORE 2 Module 15
722
Lesson 2
Describing End Behavior of a Decay Function
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U6M15L2 722
Math Background
The end behavior of a function f(x) is a description of what happens to a function f(x) as x increases or decreases without bound. The value of an exponential growth function f(x) = ab x, where b > 1 and a > 0, increases without bound as x increases without bound, and approaches 0 as x decreases without bound. The value of an exponential decay function f(x) = ab x, where 0 < b < 1 and a > 0, approaches 0 as x increases without bound, and increases without bound as x decreases without bound. Such end behavior can also be described using the notation f(x) → ∞ as x → ∞ and f(x) → 0 as x → ∞, where the symbol ∞ represents infinity.
19/04/14 10:54 AM
QUESTIONING STRATEGIES Before you graph it, how can you tell that the x function f(x) = 500(0.8) will decrease as x increases? A positive number is being multiplied x times by a number between 0 and 1, so it will get smaller as x increases.
Modeling Exponential Growth and Decay
722
Recall that a function of the form y = ab x represents exponential growth when a > 0 and b > 1. If b is replaced by t 1 + r and x is replaced by t, then the function is the exponential growth model y = a(1 + r) , where a is the initial amount, the base (1 + r) is the growth factor, r is the growth rate, and t is the time interval. The value of the model increases with time.
What does it mean for a function to approach 0 as x increases without bound? Is 0 a value of the function? The value of the function gets closer and closer to 0 as x increases, but it never reaches 0.
Example 1
EXPLAIN 1
A painting is sold for $1800, and its value increases by 11% each year after it is sold. Find the value of the painting in 30 years.
y = a(1 + r)
t
= 1800(1 + 0.11) = 1800(1.11)
QUESTIONING STRATEGIES
t
t
Find the value in 30 years. y = 1800(1.11) = 1800(1.11)
t
30
≈ 41,206.13 After 30 years, the painting will be worth approximately $41,206.13. Create a table of values to graph the function. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Tetra Images/Corbis
Why doesn’t the graph of an exponential growth model intersect the y-axis at the origin? When x = 0, y is the initial value of the increasing quantity.
Write an exponential growth function for each situation. Graph each function and state its domain, range and an asymptote. What does the y-intercept represent in the context of the problem?
Write the exponential growth function for this situation.
Modeling Exponential Growth
How do you use a percent increase amount when writing an exponential growth equation? Give an example. Convert the percent increase to a decimal and add it to 1 to find the number that is raised to a power. Possible answer: For a 4% increase, the decimal 1.04 will be raised to a power.
Modeling Exponential Growth
Explain 1
QUESTIONING STRATEGIES
t
y
0
1800.00
(t, y)
y
(0, 1800)
44,000
8
4148.20
(8, 4148.20)
16
9559.60
(16, 9559.60)
24
22,030.00
(24, 22,030)
32
50,770.00
(32, 50,770.00)
Determine the domain, range and an asymptote of the function.
(32, 50,770.00)
33,000 22,000
(24, 22,030)
11,000 (0, 1800) (16, 9559.60) t (8, 4148.20) 0 8 16 24 32
The domain is the set of real numbers t such that t ≥ 0. The range is the set of real numbers y such that y ≥ 1800. An asymptote for the function is y = 0. The y-intercept is the value of y when t = 0, which is the value of the painting when it was sold.
Module 15
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Lesson 2
COLLABORATIVE LEARNING IN1_MNLESE389762_U6M15L2 723
Small Group Activity Divide students into groups of three or four. Have students research several banks, either locally or online, for interest rates on savings accounts, money market accounts, and CDs. Have them work as a group to develop exponential growth models for each type of account at each bank on an initial deposit of $500. Ask students to make a chart comparing the rates and potential earnings after 5 years for the different types of accounts at each bank.
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Lesson 15.2
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B
A baseball trading card is sold for $2, and its value increases by 8% each year after it is sold. Find the value of the baseball trading card in 10 years. Write the exponential growth function for this situation. y = a(1 + r)
t
(1 + 0.08 ) 2 ( 1.08 )
= 2 =
t
t
Find the value in 10 years. y = a(1 + r)
t
( 1.08 ) 2 ( 1.08 )
= 2 =
t
10
≈ 4.32
After 10 years, the baseball trading card will be worth approximately $ 4.32 million dollars . Create a table of values to graph the function.
t
(t, y)
y
6
(0, 2)
0
2
3
2.5194
6
3.1737
9
3.998
12
5.0363
4.5
(3, 2.5194) (6, 3.1737) (9, 3.998) (12, 5.0363)
3 1.5 0
y (12, 5.0363) (9, 3.998) (6, 3.1737) (3, 2.5194) (0, 2) 3
t 6
9
12
Determine the domain, range, and an asymptote of the function.
The range is the set of real numbers y such that y ≥ 2 . An asymptote for the function is
y=0
.
The y-intercept is the value of y when t = 0, which is the
value of the card when it was sold
.
Reflect
8.
Find a recursive rule that models the exponential growth of y = 1800( 1.11 )t. a 1 = 1800, a n = 1.11a n-1
9.
Find a recursive rule that models the exponential growth of y = 2(1.08) . a 1 = 2, a n = 1.08a n-1
Module 15
© Houghton Mifflin Harcourt Publishing Company
The domain is the set of real numbers t such that t ≥ 0 .
t
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Lesson 2
DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U6M15L2 724
Critical Thinking
19/04/14 10:54 AM
To help students understand the difference between exponential growth and linear growth, point out that in exponential growth or decay, the amount added or subtracted in each time period is proportional to the amount already present. For exponential growth, this means that as the amount becomes greater, the amount of increase in each time period also becomes greater. Contrast this to linear growth, in which the amount of increase remains constant.
Modeling Exponential Growth and Decay
724
Your Turn
EXPLAIN 2
10. Write and graph an exponential growth function, and state the domain and range. Tell what the y-intercept represents. Sara sold a coin for $3, and its value increases by 2% each year after it is sold. Find the value of the coin in 8 years. t 8 y = a (1 + r ) y = 3(1.02)
Modeling Exponential Decay
= 3(1.02)
≈ 3.51
t
QUESTIONING STRATEGIES
4 3
y (6, 3.38) (2, 3.12) (4, 3.25) (8, 3.52) (0, 3)
2 1
After 8 years, the coin will be worth approximately $3.51.
What is the domain for a function that represents the exponential growth or decay of a population? Explain. The domain is all non-negative real numbers, because the model applies to all future times (positive values of t) but not past times (negative values of t).
The domain is the set of real numbers t such that t ≥ 0.
t 0
2
4
6
8
The range is the set of real numbers y such that y ≥ 3.
The y-intercept is the value of y when t = 0, which is the value of the coin when it was sold.
Modeling Exponential Decay
Explain 2
Recall that a function of the form y = ab x represents exponential decay when a > 0 and 0 < b < 1. If b is replaced t by 1 - r and x is replaced by t, then the function is the exponential decay model y = a(1 - r) , where a is the initial amount, the base (1 - r) is the decay factor, r is the decay rate, and t is the time interval.
What is the range for a function describing the exponential growth or decay of a population? For a growth function, the range consists of all numbers greater than or equal to the starting value of the population. For a decay function, the range includes all numbers between the starting value and 0.
Example 2
Write an exponential decay function for each situation. Graph each function and state its domain and range. What does the y-intercept represent in the context of the problem?
The population of a town is decreasing at a rate of 3% per year. In 2005, there were 1600 people. Find the population in 2013. Write the exponential decay function for this situation. y = a(1 - r)
t
= 1600(1 - 0.03) © Houghton Mifflin Harcourt Publishing Company
= 1600(0.97)
t
t
Find the value in 8 years. y = 1600(0.97)
t
= 1600(0.97)
8
≈ 1254 After 8 years, the town’s population will be about 1254 people.
Module 15
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Lesson 2
LANGUAGE SUPPORT IN1_MNLESE389762_U6M15L2 725
Connect Vocabulary Remind students that exponential growth refers to an increasing function and exponential decay refers to a decreasing function. While students are probably familiar with the word growth, they may be less familiar with decay. Explain that, in everyday use, it can mean rot, or loss of strength and health. Have students list key words that can indicate growth or decay in descriptions of real-world situations. Examples of key words for growth include increases, goes up, rises, gains. Examples of key words for decay include decreases, goes down, falls, loses value, declines, depreciates.
725
Lesson 15.2
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Create a table of values to graph the function.
(t, y)
t
y
0
1600
8
1254
16
983
24
770
32
604
y (0, 1600)
1600
(8, 1254)
1200
(0, 1600)
(16, 983)
800
(8, 1254)
(24, 770) (32, 604)
400
(16, 983)
t
(24, 770)
0
8
16
24
32
(32, 604)
Determine the domain and range of the function. The domain is the set of real numbers t such that t ≥ 0 The range is the set of real numbers y such that 0 ≤ y ≤ 1600. The y-intercept is the value of y when t = 0, the number of people before it started to lose population.
B
The value of a car is depreciating at a rate of 5% per year. In 2010, the car was worth $32,000. Find the worth of the car in 2013. Write the exponential decay function for this situation. y = a(1 - r)
t
( ) 32,000 ( 0.95 )
t
= 32,000 1 - 0.05 t
=
Find the value in 3 years. y = a(1 + r)
t
( 0.95 ) 32,000 ( 0.95 ) ≈
= 32,000
3
27,436
© Houghton Mifflin Harcourt Publishing Company
=
t
After 3 years, the car’s value will be $ 27,436 . Create a table of values to graph the function.
t
y
0
32,000
1
30,400
2
28,880
3
27,346
(t, y)
(0, 32,000) (1, 30,400) (2, 28,880) (3, 27,346)
Determine the domain and range of the function.
y (1, 30,400) 32,000 (3, 28,346) (0, 32,000) (2, 28,880) 24,000 16,000 8,000 t 0
1
2
3
4
The domain is the set of real numbers t such that t ≥ 0 . The range is the set of real numbers y such that 0 ≤ y ≤ 32,000 .
The y-intercept, 32,000, is the value of y when t = 0, the original value of the car.
Module 15
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Lesson 2
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Modeling Exponential Growth and Decay
726
Reflect
EXPLAIN 3
11. Find a recursive rule that models the exponential decay of y = 1600(0.97) . a 1 = 1,600, a n = 0.97a n-1 t
Comparing Exponential Growth and Decay
12. Find a recursive rule that models the exponential decay of y = 32,000(0.95) . a 1 = 32,000, a n = 0.95a n-1 t
Your Turn
AVOID COMMON ERRORS
13. The value of a boat is depreciating at a rate of 9% per year. In 2006, the boat was worth $17,800. Find the worth of the boat in 2013. Write an exponential decay function for this situation. Graph the function and state its domain and range. What does the y-intercept represent in the context of the problem?
Some students may forget to convert the percent growth rate to decimal form. Remind them that growth rate must be written as a decimal because the percent sign means “parts out of 100.”
y = a(1 - r) = 17,800(0.91) t
t
y 20,000 (0, 17,800) (2, 14,740) 15,000
7
After 7 years, the boat will be worth approximately $9,198.35. The domain is the set of real numbers t such that t ≥ 0. The range is the set of real numbers y such that 0 ≤ y ≤ 17,800.
QUESTIONING STRATEGIES
(8, 8370.5)
5,000
y = 17,800(0.91) ≈ 9198.35
(6, 10,108)
(4, 12,206)
10,000
t 0
2
4
6
8
The y-intercept is 17,800, the value of y when t = 0, which is the original value of the boat.
When you graph two functions on the same coordinate grid, what does the intersection point of the graphs represent? the time when the two functions are equal in value
Explain 3
Comparing Exponential Growth and Decay
Graphs can be used to describe and compare exponential growth and exponential decay models over time. Example 3
© Houghton Mifflin Harcourt Publishing Company
Use the graphs provided to write the equations of the functions. Then describe and compare the behaviors of both functions.
The graph shows the value of two different shares of stock over the period of 4 years since they were purchased. The values have been changing exponentially.
16 12
y (0, 16) Stock A (1, 12)
8
The graph for Stock A shows that the value of the stock is decreasing as time increases.
4
The initial value, when t = 0, is 16. The value when t = 1 is 12. Since 12 ÷ 16 = 0.75, the function that represents the value of Stock A after t t years is A(t) = 16(0.75) . A(t) is an exponential decay function.
0
Stock B (0, 2) 1
(1, 3) 2
x 3
4
The graph for Stock B shows that the value of the stock is increasing as time increases. The initial value, when t = 0, is 2. The value when t = 1 is 3. Since 3 ÷ 2 = 1.5, the t function that represents the value of Stock B after t years is B(t) = 2(1.5) . B(t) is an exponential growth function. The value of Stock A is going down over time. The value of Stock B is going up over time. The initial value of Stock A is greater than the initial value of Stock B. However, after about 3 years, the value of Stock B becomes greater than the value of Stock A. Module 15
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Lesson 15.2
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B
The graph shows the value of two different shares of stocks over the period of 4 years since they were purchased. The values have been changing exponentially. The graph for Stock A shows that the value of the stock is
decreasing
y 100 (0, 100) Stock A 75
as time increases.
The initial value, when t = 0, is 100 . The value when t = 1 is 50 . Since 50 ÷ 100 = 0.5 , the function that
t
represents the value of Stock A after t years is A(t) = 100 0.5 . A(t) is an exponential decay function.
25 (0, 1.5) 0
exponential growth and decay, ensure that students understand the meaning of the real-world values on the graph. Point out that the independent variable represents time, while the meaning of the dependent variable can vary with the situation. Remind students to label the axes of their graphs with the correct quantities and units. Students should also recognize that the y-intercept is the value of the independent variable at time t = 0, which may be called its initial value or starting value.
(1, 50)
50
( )
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 When modeling real-world examples of Stock B
(1, 3) 1
2
3
4
The graph for Stock B shows that the value of the stock is increasing as time increases. The initial value, when t = 0, is 1.5 . The value when t = 1 is 3 . Since 3 ÷ 100 = 0.5 , the function that represents the value of Stock B after t years is B(t) = 1.5 ( 2 )t. B(t) is an exponential
growth function. The value of Stock A is going down over time. The value of Stock B is going up over time. The initial value of Stock A is greater than the initial value of Stock B. However, after about
3 years,
the value of Stock B becomes greater than the value of Stock A. Reflect
14. Discussion In the function B(t) = 1.5(2) , is it likely that the value of B can be accurately predicted in 50 years? The exponential function grows much more quickly than any stock can reasonably grow, it t
is unlikely that the value of B can be accurately predicted in 50 years by using the function.
15. The graph shows the value of two different shares of stocks over the period of 4 years since they were purchased. The values have been changing exponentially. Use the graphs provided to write the equations of the functions. Then describe and compare the behaviors of both functions. y 150 (0, 150) Stock B: Stock A: Stock A 100 t(0) = 5 t(0) = 150 (1, 45) t(1) = 45 t(1) = 15 50 Stock B (1, 15) 45 ÷ 150 = .3 15 ÷ 5 = 3 t (0, 5) t t 0 1 2 3 ) ) ( ( ) ) ( ( A t = 150 0.3 B t =5 3
© Houghton Mifflin Harcourt Publishing Company
Your Turn
The value of Stock A is going down over time. The value of Stock B is going up over time. The initial value of Stock A is greater than the initial value of Stock B. However, after about 1.5 years, the value of Stock B becomes greater than the value of Stock A. Module 15
IN1_MNLESE389762_U6M15L2 728
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Lesson 2
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Modeling Exponential Growth and Decay
728
Elaborate
ELABORATE
16. If b > 1 in a function of the form y = ab x, is the function an example of exponential growth or an example of exponential decay? If b >1, the function is an example of exponential growth.
QUESTIONING STRATEGIES
17. What is an asymptote of the function y = 35( 1.1 )x ? x An asymptote of the function y = 35(1.1) is y = 0.
Describe three real-world situations that can be described by exponential growth or exponential decay functions. Possible answers: interest earned on an investment, population growth or decline, radioactive decay
18. Essential Question Check-In What equation should be used when modeling an exponential function that models a decrease in a quantity over time? When modeling an exponential function that models a decrease in a quantity over time,
the equation y = a(1 - r) should be used. t
SUMMARIZE How do you write an exponential growth or decay function? The formula for growth is t y = a(1 + r) , and the formula for decay is t y = a(1 - r) , where y represents the final amount, a represents the original amount, r represents the rate of growth expressed as a decimal, and t represents time.
Evaluate: Homework and Practice Graph the function on a graphing calculator, and state its domain, range, end behavior, and an asymptote. 1.
ƒ(x) = 300(1.16)
© Houghton Mifflin Harcourt Publishing Company
ƒ(x) = 65(1.64)
x
ƒ(x) = 800(0.85)
x
Domain: {x| - ∞ < x < ∞} Range: {y| y > 0} End behavior: As x → -∞, y → ∞ and as x → ∞, y → 0 Asymptote: y = 0 4.
Domain: {x| - ∞ < x < ∞} Range:{y| y > 0} End behavior: As x → -∞, y → 0 and as x → ∞, y → ∞ Asymptote: y = 0
ƒ(x) = 57(0.77)
x
Domain: {x| - ∞ < x < ∞} Range: {y| y > 0} End behavior: As x → -∞, y → ∞ and as x → ∞, y → 0 Asymptote: y = 0
Write an exponential function to model each situation. Then find the value of the function after the given amount of time. 5.
Annual sales for a company are $155,000 and increases at a rate of 8% per year for 9 years.
6.
t
= 69(0.85)
y = 155,000(1.08)
9
IN1_MNLESE389762_U6M15L2 729
The value of a textbook is $69 and decreases at a rate of 15% per year for 11 years.
y = 69(0.85)
y = 155,000(1.08)
t
11
= 155,000(1.999)
= 69(0.167)
= $309,845.72
= $11.52
Module 15
Lesson 15.2
2.
Domain: {x| - ∞ < x < ∞} Range: {y| y > 0} End behavior: As x → -∞, y → 0 and as x → ∞, y → ∞ Asymptote: y = 0 3.
729
x
• Online Homework • Hints and Help • Extra Practice
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Lesson 2
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7.
A new savings account is opened with $300 and gains 3.1% yearly for 5 years.
8.
y = 300(1.031)
y = 7800(0.92)
= 300(1.16)
= 7800(0.61)
t
= 300(1.031)
= 7800(0.92)
5
= $349.47
9.
The value of a car is $7800 and decreases at a rate of 8% yearly for 6 years.
EVALUATE
t
6
= $4729.57
The starting salary at a construction company is fixed at $55,000 and increases at a rate of 1.8% yearly for 4 years.
y = 55,000(1.018)
ASSIGNMENT GUIDE
t
= 55,000(1.018)
4
= 55,000(1.074) = $59,068.21
10. The value of a piece of fine jewelry is $280 and decreases at a rate of 3% yearly for 7 years.
y = 280(0.97) = 280(0.97)
11. The population of a town is 24,000 and is increasing at a rate of 6% per year for 3 years.
y = 24,000(1.06)
t
= 24,000(1.06)
7
= 280(0.808)
t
= 28,584
12. The value of a new stadium is $3.4 million and decreases at a rate of 2.39% yearly for 10 years.
y = 3.4(0.9761)
t
10
= 3.4(0.785)
= $2.67 million
Write an exponential function for each situation. Graph each function and state its domain and range. Determine what the y-intercept represents in the context of the problem.
200,000
13. The value of a boat is depreciating at a rate of 7% per year. In 2004, the boat was worth $192,000. Find the worth of the boat in 2013.
100,000
y = a(1 - r) = 192,000(0.93) t
y (0, 192,000)
150,000
t
50,000
y = 192,000(0.93) ≈ 99,918,93 9
(4, 143,626) (8, 107,440) (12, 80,370) t
After 9 years, the boat will be worth approximately $99,918.93. Domain: {x|0 < x < ∞} Range: {y| y > 0}
0
4
8
12
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©CoolKengzz/Shutterstock
= 3.4(0.9761)
Exercise
IN1_MNLESE389762_U6M15L2 730
COMMON CORE
Mathematical Practices
1–4
1 Recall of Information
MP.5 Using Tools
5–12
1 Recall of Information
MP.4 Modeling
13–20
2 Skills/Concepts
MP.6 Precision
21–24
2 Skills/Concepts
MP.5 Using Tools
1 Recall of Information
MP.5 Using Tools
3 Strategic Thinking
MP.3 Logic
25 26–28
Exercises 1, 3
Explore 2 Describing End Behavior of a Decay Function
Exercises 2, 4
Example 1 Modeling Exponential Growth
Exercises 5, 7, 9, 11, 14, 16, 18, 20, 25–26
Example 2 Modeling Exponential Decay
Exercises 6, 8, 10, 12–13, 15, 17, 19, 27–28
Example 3 Comparing Exponential Growth and Decay
Exercises 21–24
results on a graphing calculator. They can enter the function in the Y = menu and then press GRAPH. Then they can press the TABLE key to get a list of x- and y-values, and can scroll up or down to see more values. This method is especially convenient when seeking the value of the function for more than one input value, because it is an alternative to repeatedly typing the expression into the calculator.
Lesson 2
730
Depth of Knowledge (D.O.K.)
Explore 1 Describing End Behavior of a Growth Function
INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Students may want to check their
The y-intercept is 192,000, the original value of the boat in 2004.
Module 15
Practice
3
= 24,000(1.191)
= $226.24
Concepts and Skills
19/04/14 10:53 AM
Modeling Exponential Growth and Decay
730
14. The value of a collectible baseball card is increasing at a rate of 0.5% per year. In 2000, the card was worth $1350. Find the worth of the card in 2013.
VISUAL CUES Focus on visual cues as you discuss how to recognize exponential growth and decay from a graph. Students should recognize that a rising line represents growth and a falling line represents decay. The steeper the line, the greater the rate of growth or decay.
y = a(1 + r) = 1350(1.005) t
t
y = 1350(1.005) ≈ 1440.43 13
After 13 years, the card will be worth approximately $1440.43. Domain: {x|0 < x < ∞} Range: {y| y > 0} The y-intercept is 1350.00, the original value of the card in 2000.
AVOID COMMON ERRORS
500 t 0
15. The value of an airplane is depreciating at a rate of 7% per year. In 2004, the airplane was worth $51.5 million. Find the worth of the airplane in 2013.
CONNECT VOCABULARY
y = a(1 - r) = 51.5(0.93) t
Students may not be familiar with the word depreciate. Explain that to depreciate is to decrease in value. One meaning of the word appreciate is the opposite of depreciate: to increase in value.
30
After 9 years, the airplane will be worth approximately $26.8 million.
20
Domain: {x|0 < x < ∞} Range: {y| y > 0}
10
The y-intercept is 51.5, the original value of the airplane in 2004.
16. The value of a movie poster is increasing at a rate of 3.5% per year. In 1990, the poster was worth $20.25. Find the worth of the poster in 2013.
y = a(1 + r) = 20.25(1.035) t
y = 20.25(1.035)
23
t
≈ 44.67
Domain: {x|0 < x < ∞} Range: {y| y > 0} The y-intercept is 20.25, the original value of the poster in 1990.
731
10
15
y (0, 51.5) (4, 38.5) (6, 33.3)
(10, 24.9)
(8, 28.8)
t 0
50
2
4
6
8
10
y
40
(15, 33.93)
30 (5, 24.05)
After 23 years, the poster will be worth approximately $44.67.
IN1_MNLESE389762_U6M15L2 731
5
40 (2, 44.5)
t
9
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©RoseOfSharon/Alamy
50
y = 51.5(0.93) ≈ 26.8
Module 15
Lesson 15.2
(15, 1454.90)
(0, 1350) (10, 1419.00) 1,000
Some students may forget to add 1 to the rate of growth in the exponential growth model. Remind students that 1 + r is the growth factor in the model.
731
(5, 1384.10) y
1,500
20 (0, 20.25)
(20, 40.29)
(10, 28.57)
10 t 0
5
10
15
20
25
Lesson 2
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17. The value of a couch is decreasing at a rate of 6.2% per year. In 2007, the couch was worth $1232. Find the worth of the couch in 2014. The y-intercept is 20.25, the value of y when t = 0, which is the original value of the poster in 1990.
y = a(1 - r) = 1232(0.938) t
1600
y
1200 (2, 1083.97) 800
t
y = 1232(0.938) ≈ 787.10 7
QUESTIONING STRATEGIES
(0, 1232.00)
The population of a town is decreasing at the rate of 1% a year. You are asked to find the population after four years. Solving the exponential growth function, you get an answer of 1248.77. What answer will you record? Explain. Round the answer to 1249 because there cannot be a fraction of a person.
(4, 953.72) (6, 839.19) (8, 738.30)
400
After 7 years, the couch will be worth approximately $787.10. Domain: {x|0 < x < ∞} Range: {y| y > 0}
0
2
4
6
8
The y-intercept is 1232, the original value of the couch in 2007. 18. The population of a town is increasing at a rate of 2.2% per year. In 2001, the town had a population of 34,567. Find the population of the town in 2018.
y = a(1 + r) = 34,567(1.022) t
50,000
y
(9, 42,046) (3, 36,899)
40,000
t
y = 34,567(1.022) ≈ 50,041
30,000 (0, 34,567)
After 17 years, the town will have about 50,041 people.
20,000
17
Domain: {x|0 < x < ∞} Range: {y| y > 0}
y = a(1 - r) = 131,000(0.946) t
t
y = 131,000(0.946) ≈ 75,194 10
150,000
30,000
20. An account is gaining value at a rate of 4.94% per year. The account held $113 in 2005. What will the bank account hold in 2017? t
y = 113(1.0494) ≈ 201.54 12
200
1
40
2
y
160 (0, 113) 120 80
The y-intercept is 113, the original amount in the account in 2005.
12
15
(4, 104,915)
t
Domain: {x|0 < x < ∞} Range: {y| y > 0}
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9
(0, 131,000) (2, 117,234)
0
After 12 years, the bank account will hold about $201.54.
732
y
6
3
4
5
(8, 166.19) (4, 137.04) (6, 150.91) (2, 124.44)
© Houghton Mifflin Harcourt Publishing Company
Domain: {x|0 < x < ∞} Range: {y| y > 0} The y-intercept is 131,000, the original value of the house in 2009.
Module 15
3
120,000 (1, 123,926) 90,000 (3, 110,903) 60,000
t
Some students may fail to write percents as decimals before applying the exponential growth model. Encourage them to write out the decimal equivalent of a growth or decay rate before adding it to 1 or subtracting it from 1 in the exponential equation.
t 0
After 10 years, the house will be worth about $75,194.
y = a(1 + r) = 113(1.0494)
AVOID COMMON ERRORS
(12, 44,882) (6, 39,388)
10,000
The y-intercept is 34,567, the original population of the town in 2001.
19. A house is losing value at a rate of 5.4% per year. In 2009, the house was worth $131,000. Find the worth of the house in 2019.
(15, 47,910)
t 0
2
4
6
8
10
Lesson 2
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Modeling Exponential Growth and Decay
732
Describe and compare each pair of functions.
PEERTOPEER DISCUSSION
21. A(t) = 13(0.6) and B(t) = 4(3.2) t
Ask students to discuss with a partner how to write an exponential growth or decay model from a verbal description of a real-world situation. They should consider which key words tell whether the situation involves growth or decay; what the starting value a is; and how to use the growth or decay rate r in the equation.
t
The value of A(t) is decreasing. The value of B(t) is increasing. The initial value of A(t) is greater than the initial value of B(t). However, after about .7 units, the value of B(t) becomes greater than the value of A(t).
22. A(t) = 9(0.4) and B(t) = 0.6(1.4) t
t
The value of A(t) is decreasing. The value of B(t) is increasing. The initial value of A(t) is greater than the initial value of B(t). However, after about 2.2 units, the value of B(t) becomes greater than the value of A(t).
23. A(t) = 547(0.32) and B(t) = 324(3) t
t
The value of A(t) is decreasing. The value of B(t) is increasing. The initial value of A(t) is greater than the initial value of B(t). However, after about .2 units, the value of B(t) becomes greater than the value of A(t).
JOURNAL
t t 24. A(t) = 2(0.6) and B(t) = 0.2(1.4)
Have students write a journal entry in which they describe how to graph an exponential growth or decay function, and how to find the equation for a real-world situation involving exponential growth or decay.
The value of A(t) is decreasing. The value of B(t) is increasing. The initial value of A(t) is greater than the initial value of B(t). However, after about 2.7 units, the value of B(t) becomes greater than the value of A(t). 25. Identify the y-intercept of each of the exponential functions. a. 3123(432,543) b. 0 c. 45(54)
x
x
(0, 3123)
d. 76(89, 047, 832)
(0, 0)
e. 1
x
(0, 76) (0, 1)
(0, 45)
© Houghton Mifflin Harcourt Publishing Company
H.O.T. Focus on Higher Order Thinking
26. Explain the Error A student was asked to find the value of a $2500 item after 4 years. The item was depreciating at a rate of 20% per year. What is wrong with the student’s work? 4 25000(0.2) $4 t t The exponential decay function is y = 2500(1 - 0.2) , or y = 2500(0.8) . The student forgot to subtract the rate of depreciation from 1 before solving. 27. Make a Conjecture The value of a certain car can be modeled by the function t y = 18000(0.76) , where t is time in years. Will the value of the function ever be 0?
The value of the function will never be 0 because the right side of the function is a product of positive numbers. Although the value can become extremely close to 0, it can never equal 0. 28. Communicate Mathematical Ideas Explain how a graph of an exponential function may resemble the graph of a linear function.
A graph of an exponential function may appear to be a linear function if only a small part of the graph is shown and the values in that part are changing slowly.
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Lesson 15.2
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Lesson 2
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Lesson Performance Task
CONNECT VOCABULARY Some students may not be familiar with the terms archeologist, artifacts, or radiometric dating. Explain that an archeologist studies prehistoric people and cultures by analyzing items that remain for us to find, such as their bones, weapons, artifacts, and tools. Artifacts are objects that the people made, such as pottery. Radiometric dating, also called radioactive dating, is a method of determining the age of an object based on the rate of decay of an element in the object.
Archeologists have several methods of determining the age of recovered artifacts. One method is radioactive dating. All matter is made of atoms. Atoms, in turn, are made of protons, neutrons, and electrons. An “element” is defined as an atom with a given number of protons. Carbon, for example, has exactly 12 protons. Carbon atoms can, however, have different numbers of neutrons. These are known as “isotopes” of carbon. Carbon-12 has 12 neutrons, carbon-13 has 13 neutrons, and carbon-14 has 14 neutrons. All carbon-based life forms contain these different isotopes of carbon. Carbon-12 and carbon-13 account for over 99% of all the carbon in living things. Carbon-14, however, accounts for approximately 1 part per trillion or 0.0000000001% of the total carbon in living things. More importantly, carbon-14 is unstable and has a half-life of approximately 5700 years. This means that, within the span of 5700 years, one-half of any amount of carbon will “decay” into another atom. In other words, if you had 10 g of carbon-14 today, only 5 g would remain after 5700 years. But, as long as an organism is living, it keeps taking in and releasing carbon-14, so the level of it in the organism, as small as it is, remains constant. Once an organism dies, however, it no longer ingests carbon-14, so the level of carbon-14 in it drops due to radioactive decay. Because we know how much carbon-14 an organism had when it was alive, as well as how long it takes for that amount to become half of what it was, you can determine the age of the organism by comparing these two values.
QUESTIONING STRATEGIES What is the growth factor (the base raised to a power) in the exponential model? Explain. It is 1 - 0.5 = 0.5, because the amount of carbon-14 decreases by half during every time period.
n
Carbon-14 Concentration In Parts Per Quadrillion
C(n) = 1000(0.5)
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©worker/ Shutterstock
Use the information presented to create a function that will model the amount of carbon-14 in a sample as a function of its age. Create the model C(n) here C is the amount of carbon-14 in parts per quadrillion (1 part per trillion is 1000 parts per quadrillion) and n is the age of the sample in half-lives. Graph the model.
1000 800 600 400 200 0
2
4
6
8
Age in Half-Lives
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EXTENSION ACTIVITY IN1_MNLESE389762_U6M15L2 734
Have students research how very large numbers such as million, billion, trillion, and quadrillion compare to each other. Then have students rewrite the model C(n) in different units and tell how the graph would change.
Students may find that since 1 trillion, or 10 12, is equal to 1000 billion, or 10 3 ∙ 10 9, 1 part per trillion is the same as 0.001 parts per billion. Therefore, the model could be C(n) = 0.001(0.5)n. The graph would differ only in that the scale of the vertical axis would be in parts per billion rather than parts per quadrillion.
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Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Modeling Exponential Growth and Decay
734
LESSON
15.3
Name
Using Exponential Regression Models
Class
Date
15.3 Using Exponential Regression Models Essential Question: How can you use exponential regression to model data? Resource Locker
Common Core Math Standards The student is expected to:
One of the reasons data is valuable is that it allows us to make predictions for values that fall outside of the data set. In order to do this, the data needs to be synthesized into a function. An exponential regression is a graphing calculator tool used to generate an exponential equation that fits data exhibiting exponential growth or decay. The statistical tools on a graphing calculator offer several possible methods for finding a regression model for a set of data. Use a graphing calculator to find the exponential regression equation that models the data provided.
Fit a function to the data ... to solve problems in the context of the data. Also S-ID.B.6b, A-CED.A.2, A-REI.D.11, F-LE.A.1c
Mathematical Practices COMMON CORE
Fitting an Exponential Function to Data
Explore
S-ID.B.6a
MP.5 Using Tools
Number of Internet Hosts
Demonstrate how to use residuals to evaluate how well an exponential regression equation fits a set of data.
Number (millions)
ENGAGE Essential Question: How can you use exponential regression to model data? You can enter the data into a graphing calculator and use the exponential regression program to generate an equation for the data. Plot and analyze the residuals to determine how well the model fits the data and whether it can be used to make predictions.
PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss why it would be better to save money in an interest-bearing savings account than to store it in a shoebox. Then preview the Lesson Performance Task.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Chad Baker/ Getty Images
Language Objective
Years since 2001
0
1
2
3
4
5
6
110
147
172
233
318
395
433
Enter the data from the table on a graphing calculator, with years since 2001 in L1. Input the number of Internet hosts in L2. Create a scatter plot of the data on the calculator. Plot the data points on the given grid.
Use the statistical calculation features of a graphing calculator to calculate the exponential regression equation for the data you entered into L1 and L2.
Number (millions)
COMMON CORE
The exponential regression function is x x f(x) = 113(1.27) or y = 113(1.27) . (Round to three significant digits.)
Graph the exponential regression equation with the data points on the calculator. Sketch a graph of the exponential regression equation on the grid with the data points that you plotted.
700 600 500 400 300 200 100 0
y
x 2
4
6
8
Years since 2001
Reflect
1.
Discussion Which parameter, a or b, represents the initial value of the function? Explain how you know. The initial value of the function is represented by a. The initial value of the function corresponds to f(x) when x = 0, and any number raised to the zero power is 1. f(x) = a(b) and f(0) = a(b) = a(1) = a x
0
So the initial value of the function is 113, which is also close to the first data point, (0,110). Module 15
be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
Lesson 3
735
gh “File info”
made throu
Date Class
al Exponenti 15.3 Using ssion Models Regre
Name
(millions Number ator, ing calcul on a graph the table er of the numb data from on Enter the in L1. Input of the data since 2001 a scatter plot with years grid. in L2. Create on the given Internet hosts the data points ator. Plot the calcul a graphing on features of ation sion equati ical calcul ential regres Use the statist the expon to calculate and L2. L1 into calculator d you entere for the data on is x sion functi (1.27) . ential regres x The expon (1.27) or y = 113 f(x) = 113 digits.) cant three signifi with the data (Round to equation ential regression the expon exponential a graph of Graph the points that ator. Sketch the calcul with the data points on on the grid equation regression d. you plotte
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Years since
2001
2
4
Harcour n Mifflin © Houghto Getty Images
you know. Explain how function? ion value of the the funct the initial l value of b, represents d by a. The initia 1. eter, a or sente power is Which param ion is repre to the zero Discussion the funct er raised l value of any numb The initia ). x = 0, and f(x) when point, (0,110 to ds data a 0 = correspon to the first ) = a(1) Lesson 3 x also close (0) = a(b which is (b) and f ion is 113, f(x) = a the funct l value of 735 So the initia
Reflect
1.
Module 15
5L3 735 62_U6M1
ESE3897
Lesson 15.3
y 700 600 500 400 300 200 100
Watch for the hardcover student edition page numbers for this lesson.
IN1_MNL
735
(millions)
Credits: Chad
Baker/
Number
IN1_MNLESE389762_U6M15L3 735
HARDCOVER PAGES 735748
Resource
l data? Locker sion to mode 6b, ential regres Also S-ID.B. t of the data. you use expon How can in the contex problems Question: ... to solve Essential to the data a function n to Data COMMON S-ID.B.6a Fit set. F-LE.A.1c CORE l Functio A-REI.D.11, e of the data A-CED.A.2, fall outsid Exponentia values that sion is a graphing tions for regres Fitting an or decay. to make predic An exponential us growth Explore ential is that it allows into a function. model for ting expon sized is valuable data exhibi finding a regression provided. to be synthe reasons data on that fits ds for s the data data needs One of the ential equati l possible metho on that model do this, the te an expon sion equati offer severa In order to to genera calculator ential regres tool used graphing the expon a calculator on find to tor cal tools ng calcula The statisti Use a graphi a set of data. et Hosts 6 of Intern 5 4 Number 3 2 1 0 395 433 2001 233 318 Years since 147 172 ) 110
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11:43 AM
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Reflect
2.
EXPLORE 1
What is the growth rate of this exponential model? The growth rate is 0.27 which corresponds to a 27% yearly increase.
Fitting an Exponential Function to Data
Plotting and Analyzing Residuals of Exponential Models
Explore 2
Recall that a residual is the difference between the actual y-value in the data set and the predicted y-value. Residuals can be used to assess how well a model fits a data set. If a model fits the data well, then the following are true.
INTEGRATE TECHNOLOGY
• The numbers of positive and negative residuals are roughly equal. • The residuals are randomly distributed about the x-axis on a residual plot. • The absolute value of the residuals is small relative to the data values.
A
Students will use a graphing calculator to find an exponential regression equation in the Explore activity.
According to the data, in 2002 there were 147,000,000 Internet hosts. Find the y-value predicted by the model. y = 113(1.27)
y = 143 The actual y-value from the data is 147 . y = ab x
B
1
AVOID COMMON ERRORS
Find the difference between the data y-value and the model’s predicted y-value.
Students may incorrectly substitute values for a, b, and r into the basic exponential equation. Explain that the variable a is the coefficient while b is the base that is being taken to the x power. Thus, if a = 3 and x b = 5, the exponential equation should be y = 3(5) . Values given by the calculator should be rounded to the same number of significant digits as the input data. Point out that r is the correlation coefficient, which gives a measure of how well the equation fits the data. It is not a value to be substituted into the equation.
data - model = 147 - 143 = 4
C
On your calculator, enter the regression equation as the rule for equation Y 1. Then view the table to find the y-values predicted by model (y m). Complete the table.
Number of Internet Hosts Predicted y-value, y m
Residual yd - ym
0
110
113
-3
1
147
143
4
2
172
182
-10
3
233
231
2
4
318
294
24
5
395
373
22
6
433
474
-41
Create a residual scatter plot by plotting the x-values and the residuals in the last column as the second coordinate.
50 40 30 20 10 0 -10 -20 -30 -40 -50
Module 15
736
© Houghton Mifflin Harcourt Publishing Company
D
x
Actual y-value, y d
EXPLORE 2 Plotting and Analyzing Residuals of Exponential Models
1 2 3 4 5 6 7 8
QUESTIONING STRATEGIES Lesson 3
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U6M15L3 736
Math Background
A simple example of an exponential function is y = 2 x, in which values for y increase as follows: ƒ(1) = 2 1 = 2; ƒ(2)= 2 2 = 4; ƒ(x) = 2 3 = 8; f(4) = 2 4 = 16; and so on. This creates a rapidly increasing pattern. A classic real-world application of an exponential function is the doubling problem. Give a person a penny on day 1 and double the amount each day. How much will the person have 30 after one month? ($0.01)(2) is more than $10 million!
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How do you find the residual for each data value? First find the predicted y-values by substituting each x-value into the exponential regression model. To find the residuals, subtract each predicted y-value from the corresponding observed y-value.
Using Exponential Regression Models
736
AVOID COMMON ERRORS
Reflect
3.
Students may mistakenly compute residuals by subtracting y d from y m rather than y m from y d. This will not change the absolute value of the residuals, but it will change the sign of the residuals. Stress that it is important to calculate all residuals in the same way so that you can determine whether the number of positive and negative residuals is about equal.
are randomly distributed about the x-axis and are small relative to the data. This suggests the model is a good fit, as does the correlation coefficient. 4.
and the model. This suggests the model is becoming less accurate for later years.
Explain 1
Modeling with Exponential Functions
Example 1
1. Make a scatter plot of the data. 2. Find an exponential regression model. 3. Graph the regression function. 4. Find the input value for which the function will reach the value given in the problem. Then, students can learn the specific steps needed to complete each task.
737
Lesson 15.3
Find an exponential regression function for the given data, and use the model to make predictions.
The table shows the population y of Middleton, where x is the number of years since the end of 2000. Suppose Middleton’s town council decides to build a new high school when its population exceeds 25,000.
Years since 2000, x
Population, y
0 1 2 3 4 5 6 7
5,005 6,010 7,203 8,700 10,521 12,420 14,982 18,010
When will the population likely exceed 25,000?
Enter the x-values into L1 and the y-values into L2 in a graphing calculator and view a scatter plot of the data.
Find the exponential regression model and the regression coefficient for the data. Plot the regression function on the scatter plot. x y = 5011(1.201) r = 0.999
Use the regression model to construct an equation in one variable to solve in order to determine the time x when the population will reach 25,000. x 25,000 = 5011(1.201)
Enter y = 25,000 as Y 2 in the graphing calculator, and find the point of intersection.
AVOID COMMON ERRORS Students may be confused by the many steps involved in making a prediction based on a set of data. Encourage them to start by listing the sequence of tasks they need to accomplish, such as the following:
Modeling with Exponential Functions
Exponential regression functions can be used to make predictions.
© Houghton Mifflin Harcourt Publishing Company
EXPLAIN 1
Look for Patterns What can you infer about the accuracy of the model as it moves further away from the initial value? Explain. The residuals get larger as x increases, indicating a growing disparity between the data
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Students may think that the points on a residual plot should follow an exponential curve because they come from an exponential function. Explain that the purpose of subtracting the predicted y-values from the actual y-values is to find how much the data deviate from the prediction. You could say that the curve is subtracted out from the data. If the model fits the data well, the residuals will be relatively small and randomly distributed about the x-axis. They will not follow a sloped line or a curve.
Multiple Representations What does the residual plot reveal about the fit of the model? Does this agree with the correlation coefficient? In the residual plot, there are roughly equal numbers of positive and negative values, which
The intersection is at about ( 8.792, 25,000 ). The population will reach 25,000 in about 9 years.
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COLLABORATIVE LEARNING IN1_MNLESE389762_U6M15L3 737
Peer-to-Peer Activity Have students work in pairs. Give each pair a data set for a real-world situation that can be modeled by an exponential function, and instruct them to take turns using a graphing calculator to complete the following steps: 1. Input the data and make a scatter plot. 2. Perform an exponential regression on the data. 3. Input the exponential regression function and graph it. 4. Make a prediction about a future value of the dependent variable. Then have students reverse roles and solve a different problem.
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B
The table shows the value of a car every year since it was purchased. The owner plans to sell the car when it reaches 10% of its original value. How long will she have owned the car when she sells it?
Age of Car, x
Value, y
0
$25,000.00
1
$21,462.50
2
$17,881.88
3
$15,506.66
4
$12,919.65
5
$11,203.56
6
$9,523.03
7
$7,934.28
8
$6,880.39
9
$5,732.52
10
$4,872.64
Find the exponential regression model and the regression coefficient for the data. Round to four significant digits. y=
25,100(0.8493)
x
r=
0.9998
Use the regression model to construct an equation in one variable to solve in order to determine the time x when the value will reach 10% of the original value.
2500 = 25,100(0.8493)
x
(14.12, 2500) The intersection is at dropped to a value of $2500 after
15
. The car will have
years.
Reflect
5.
QUESTIONING STRATEGIES Why is it useful to plot the regression function on the same grid as the scatter plot? It allows you to see how closely the model fits the data. How do you use the graph of a regression function to predict when a variable will reach a given value? On the same grid where you graphed the regression function, graph a constant function equal to the given value. The point where the two graphs intersect shows the x-value at which the regression function reaches that value.
Use the regression model to predict the population at the end of 2015 and at the end of 2030. Round to four significant digits. Which prediction is likely to be more accurate? Explain your reasoning. 77,770 people; 1,208,000 people; The 2015 prediction is likely to be more accurate
because the trend of the data is more likely to continue for a short time than for a long time. 6.
During what year does the population reach 25,000? Explain your reasoning. 2009; The graphs intersect when x ≈ 8.792. Since 8 years from the end of 2000 is the end
of 2008, 8.792 years from the end of 2000 is in the latter half of 2009. 7.
design, and construction of the school must begin.
Module 15
© Houghton Mifflin Harcourt Publishing Company
Suppose the town will need a new high school already in place when the population reaches 25,000. How will the prediction above help the town make plans? The town can use the prediction to work backward and decide when plans for funding,
Lesson 3
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DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U6M15L3 738
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Modeling Students may have difficulty performing an exponential regression. Encourage them to practice the steps by first solving a simpler problem. For example, have them find an exponential regression for the values in the table: x y
0 10
1 20
2 40
Students should obtain the regression equation y = 10(2) .
3 80
x
Using Exponential Regression Models
738
Your Turn
ELABORATE
Create a model from the table of values and answer the questions. 8.
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Draw students’ attention to the way in which
Use the model to predict the census results from 2010 and compare the estimate to 6,392,017, the actual population according to the 2010 national census.
f(x) = 135.9(1.037)
they interpret a regression model to make predictions. Stress the importance of comparing the model with actual values instead of assuming that the model is a good fit for the data. Explain that a model may be more accurate for years closer to the given data than for years farther out.
SUMMARIZE
9.
Years Since 1900 (x)
Population, (y)
0
123
10
204
20
334
f(110) = 7,394
30
436
The model is significantly off from the actual value.
40
499
50
750
60
1302
70
1771
80
2718
90
3665
100
5131
x
The table shows the population of box turtles in a Tennessee wildlife park over a period of 5 years.
Year, (x)
Population, (y)
Use the model to predict the number of box turtles in the sixth year.
1
21
2
27
f(x) = 17.51(1.230)
3
33
4
41
5
48
x
f(6) = 61
© Houghton Mifflin Harcourt Publishing Company
How do you model changes in population using an exponential function? Graph the data on a graphing calculator and perform exponential regression to generate an exponential function, then compute residuals to check whether the model is a good fit. If it is, you can use the exponential regression model to make predictions about the population.
The table shows the population of Arizona (in thousands) in each census from 1900–2000.
Elaborate 10. What does a pattern in the plot of the regression data indicate? A pattern indicates that the model is not a good one. The relationship between the
dependent and independent variables is probably not exponential. 11. Discussion While it is typically the best model for population growth, what are some factors that cause population growth to deviate from the exponential format? The model does not take into account factors that might cause surges or large drops
in population. Essential Question Check-In What do the variables a and b represent in the regression equation ƒ(x) = ab x? The variable a represents the initial quantity, or the quantity at x = 0, and b represents the growth rate of the item being modeled.
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Lesson 3
LANGUAGE SUPPORT IN1_MNLESE389762_U6M15L3 739
Connect Vocabulary In the guidelines for how to evaluate residuals to determine how well a model fits the data, we see the phrase roughly equal, which means almost equal. Explain to students that when we say there are roughly equal numbers of positive values and negative values, we mean that there are close to the same number of positive values and negative values. The numbers are close to equal, but are not equal.
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Lesson 15.3
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Evaluate: Homework and Practice 1.
EVALUATE
The concentration of ibuprofen in a person’s blood was plotted each hour. An exponential model fit the data with a = 400 and b = 0.71. Interpret these parameters.
• Online Homework • Hints and Help • Extra Practice
The initial amount of ibuprofen was approximately 400 units. 71% of the previous hour’s amount of ibuprofen remains after each hour. 2.
The table shows the temperature of a pizza over three-minute intervals after it is removed from the oven. a. Find an exponential regression function for the data.
f(t) = 433(0.935)
x
Time
Temperature
0
450
3
350
6
290
9
230
12
190
15
150
18
130
21
110
ASSIGNMENT GUIDE
b. Complete the table to calculate the residuals. Plot the residuals on the scatter plot.
Temperature
Predicted Temperature
Residual
0
450
433
17
3
350
354
-4
6
290
289
1
9
230
236
-6
12
190
193
-3
15
150
157
-7
18
130
129
1
21
110
105
5
16 14 12 10 8 6 4 2 0 -2 -4 -6 -8 -10 -12 -14 -16
3 6 9 12 15 18 21
Graph y = 70 and the exponential regression function together with the graphing calculator, and find the intersection to predict how long it will take the pizza to cool down to 70° F.
c.
Practice
Explore 1 Fitting an Exponential Function to Data
Exercise 1
Explore 2 Plotting and Analyzing Residuals of Exponential Models
Exercise 2
Example 1 Modeling with Exponential Functions
Exercises 3–18
INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Spend time going over the steps for
© Houghton Mifflin Harcourt Publishing Company
Time
Concepts and Skills
entering given data in a graphing calculator, finding an exponential function that models the data, and making a prediction based on the model. Ask students to explain the purpose of each step.
It will take a little over 27 minutes for the pizza to cool to 70° F.
Module 15
Exercise
IN1_MNLESE389762_U6M15L3 740
Lesson 3
740
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1 Recall of Information
MP.2 Reasoning
2 Skills/Concepts
MP.5 Using Tools
14
3 Strategic Thinking
MP.5 Using Tools
15
2 Skills/Concepts
MP.2 Reasoning
3 Strategic Thinking
MP.3 Logic
1 2–13
16–18
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Using Exponential Regression Models
740
For Exercises 3 and 4, use a graphing calculator to calculate the exponential regression equation, and use it to solve the problem. 3.
The table shows the monthly membership in an online gaming club. When will there be more than 3000 members?
Month
Membership
0
2100
1
2163
2
2199
3
2249
4
2285
5
2329
6
2376
7
2415
8
2464
9
2514
10
2576
The club will have 3000 members in about 18 months. The table below shows a set of data that can be modeled with an exponential function. When will y be 6000?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Kidstock/ Blend Images/Corbis
4.
x
y
0
15
1
22
2
34
3
50
4
75
5
113
6
170
7
258
8
388
9
575
10
857
When x is about 14.77, y will be 6000.
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Lesson 3
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5.
A researcher is conducting an experiment on the rate that caffeine is eliminated from the body. Three volunteers are given four 8-ounce servings of coffee and asked to consume it as quickly as possible. The researchers then tested the caffeine concentration in each volunteer’s blood every 20 minutes for 4 hours to determine the rate of elimination. The table gives the results in milligrams for the three volunteers.
Time (min)
Student A (mg)
Student B (mg)
Student C (mg)
0
400
400
400
20
383
374
387
40
365
357
370
60
349
341
353
80
333
326
337
100
318
311
322
120
304
297
308
140
290
284
294
160
277
271
281
180
264
259
269
200
252
247
257
220
241
236
246
240
230
225
235
Find the hourly rate at which each student metabolizes caffeine and the time when each student will have 10 mg of caffeine in the blood.
C A(t) = 400.7(0.8705) C C(t) = 403.2(0.8738)
t t
Student A metabolizes caffeine at a rate of 12.95% per hour and will have 10 mg of caffeine in the blood after approximately 26.2 hours. Student B metabolizes caffeine at a rate of 13.09% per hour and will have 10 mg of caffeine in the blood after approximately 26.6 hours. Student C metabolizes caffeine at a rate of 12.62% per hour and will have 10 mg of caffeine in the blood after approximately 27.4 hours.
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C B(t) = 394.0(0.8691)
t
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Using Exponential Regression Models
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6.
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Ask students to think of applications of
The population of Boston, MA in thousands of people is given in the table below.
exponential functions they may have seen in their reading or studied in another course. Point out that some real-world situations can be modeled by linear functions, some can be modeled by exponential functions, and others can be modeled by more complicated functions that they have not yet studied.
1990
572
2001
602
1991
561
2002
608
1992
552
2003
608
1993
552
2004
607
1994
551
2005
610
1995
558
2006
612
1996
556
2007
623
1997
556
2008
637
1998
555
2009
645
1999
555
2010
618
2000
590
2011
625
Find a model for the population of Boston as a function of years since 1990 using the even years and a model using the odd years. Compare the models.
P e(t) = 546.67(1.007)
p o(t) = 539.61(1.008)
t t
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Tony Tremblay/E+/Getty Images
Both functions are increasing, have approximately the same rate of increase, and a similar P-intercept. They differ in exact value. Find an exponential model for the radioactive decay of the given isotope. 7.
Nobelium-253
8.
Minutes
Mass (grams)
Weeks
Mass (ounces)
0
10,000.00
0
200.00
1
6651.56
1
83.97
2
4424.33
2
35.26
3
2942.87
3
14.80
4
1957.47
4
6.22
5
1302.02
5
2.61
6
866.05
6
1.10
7
576.06
7
0.46
8
383.17
8
0.19
9
254.87
9
0.08
10
169.53
10
0.03
y = 10,000(0.665) Module 15
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y = 145(0.592)
x
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Find an exponential model for the data in the given table. 9.
x
10.
COGNITIVE STRATEGIES x
y
0
7
0
2.6
1
10.86
1
3.91
2
16.86
2
5.88
3
26.16
3
8.85
4
40.60
4
13.31
5
63
5
20.01
6
97.77
6
30.1
7
151.72
7
45.27
8
235.44
8
68.10
9
365.37
9
102.42
10
567
10
154.05
y = 7(1.552) 11.
y
y = 2.6(1.504)
x
12.
x
y
x
y
0
11
0
4
1
11.1
1
7.36
2
11.21
2
13.54
3
11.32
3
24.92
4
11.43
4
45.85
5
11.53
5
84.36
6
11.64
6
155.23
7
11.76
7
285.62
8
11.87
8
525.54
9
11.98
9
966.99
10
12.09
10
1779.26
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x
y = 4(1.840)
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x
y = 11(1.010)
Students often confuse the parameters in an exponential function. Suggest this way of thinking about them: a is the first letter of the alphabet, so it is the first y-value, and b stands for base, so it is the base for the exponent.
x
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13. The yearly profits of Company A are shown in the table. Use the information given to find a function P(t) that models the yearly profits P of the company as a function of t, the number of years since 1995.
MULTIPLE REPRESENTATIONS Some students may see an exponential equation such x as ƒ(x) = (135.9)(1.037) as exceedingly abstract. To bring the calculation down to a concrete level, do it “by hand” for ƒ(10). 10 ƒ(10) = (135.9)(1.037) = 135.9 × 1.037 × 1.037 × 1.037 × 1.037 × 1.037 × 1.037 × 1.037 × 1.037 × 1.037 × 1.037 = 195.437. Students can use a graphing calculator to confirm that the answer is the same either way.
Year
Profit (millions)
1995
5.00
1996
5.15
1997
5.30
1998
5.47
1999
5.63
2000
5.79
2001
5.97
2002
6.15
2003
6.33
2004
6.52
2005
6.71
P(t) = 5.00(1.03)
t
14. Find the Error A student is doing homework and comes to the following question.
Year
Balance
2001
$200.00
2002
$210.00
2003
$220.50
2004
$231.53
The student performs exponential regression on the data and compares the result with the answer in the back of the text.
2005
$243.10
2006
$255.26
The text gives the solution as b(t) = 200(1.05) , but the student’s model t is b(t) = 190.48(1.05) . Find the error in the student’s calculations or explain why the model is correct.
2007
$268.02
2008
$281.42
2009
$295.49
2010
$310.27
© Houghton Mifflin Harcourt Publishing Company
The table shows the balance in a student’s savings account for 10 years. The student hasn’t deposited or withdrawn any money over the time period. Find the exponential model for the student’s balance as a function of time.
t
The model is correct. The student’s model gives the balance b in the account as a function of the number of years since 2000 and the text gives the model as a function of years since 2001.
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15. Determine whether each of the following represents an increasing exponential function, a decreasing exponential function, or a non-exponential function. Select the correct answer for each part. 1t5 a. ƒ(t)= _ Increasing Decreasing Non-exponential 2 exponential exponential b. y 8
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Explain that businesses use exponential equations to compute mortgages. Typically a monthly mortgage payment is calculated using the formula
4 x -8
-4
0
4
8
-4
Increasing exponential
Decreasing exponential
Non-exponential
L⎡c(1 + c n)⎤
⎦ ⎣ ___________ , where L is the amount borrowed, ⎡⎣(1 + c) n - 1⎤⎦
-8
c.
8
x -8
-4
c is the monthly interest charge (for example, 6% ÷ 12 months = 0.005), and n is the number of months. Have students work together to calculate the monthly payment on a loan of $100,000 for a 30-year term (n = 360 months), using the monthly interest above. They should find that the monthly payment is $502.50.
y
0
4
8
Increasing exponential
Decreasing exponential
Non-exponential
Increasing exponential
Decreasing exponential
Non-exponential
Increasing exponential
Decreasing exponential
Non-exponential
Increasing exponential
Decreasing exponential
Non-exponential
-4 -8
d. ƒ(x) = 20(0.85)
e. ƒ(x) = 7(1.16) f.
x
VISUAL CUES
y
4 x -8
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When discussing how to calculate residuals, provide a visual connection by displaying the data table on an overhead projector and drawing arrows from each predicted y-value to the corresponding actual y-value. Then show the differences under the arrows.
© Houghton Mifflin Harcourt Publishing Company
8
x
4
8
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JOURNAL
H.O.T. Focus on Higher Order Thinking
Have students list the steps for making a prediction based on population data. They should include using a calculator to fit a function to the data, graphing the function, checking whether it is a good fit for the data, and using the function to make predictions.
16. Critique Reasoning The absolute values of the residuals in Mark’s regression model are less than 20. Working on a different data set, Sandy obtained residuals in the hundreds. This led Mark to conclude his data is a better fit than Sandy’s. Explain why Mark is wrong to base his assessment of their regression models on the values of the residuals.
It is the size of the residual relative to that of the data that is important. Sandy’s data values may have been quite large relative to the data values Mark was using.
17. Make a Conjecture When Chris used exponential regression on Arizona population data, he obtained the following results: a = 186, b = 1.026, r = 0.813. When he reviewed the data in his lists, he found he had entered a number incorrectly. Is it more likely that his error was in entering the last population value too high or too low? Justify your reasoning.
too low; The growth rate is too small and the initial value is too large. Entering a lower value for the last population value would have caused the modeled curve to shift “down” at the end and thus “up” at the beginning.
18. Draw Conclusions Madelyn has recorded the number of bacteria on her growth plate every hour for 3 hours. She finds that a linear model fits her data better than the expected exponential model. What should she do to improve her model?
© Houghton Mifflin Harcourt Publishing Company
She needs to gather data for longer than 3 hours. Exponential functions can appear linear, especially in the beginning.
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Lesson Performance Task
QUESTIONING STRATEGIES How can you find the initial amount in the savings account from the exponential regression equation? The initial amount is approximately the value of a in the equation y = ab x.
A student with an interest-bearing savings account reports the yearly balance in her account in the following table.
Years Since Opening the Account
Balance
1
$6152.42
2
$6305.22
3
$6459.59
4
$6626.83
5
$6793.00
6
$6965.05
7
$7148.27
8
$7322.20
9
$7505.25
10
$7710.33
11
$7906.32
12
$8092.77
How can you find the interest rate from the exponential regression equation y = ab x? Explain. Since this is an exponential growth model, we know that the base b for the exponent is equal to 1 + r, where r is the interest rate expressed as a decimal. So, to express the interest rate as a percent, find 100(b – 1).
TECHNOLOGY When students graph the data from the table, depending on the settings used, they may notice that the points appear to be nearly linear. If the window settings are changed so that the x-axis includes a greater range of values, it will become more evident that that the points follow an increasing curve. Graphing a horizontal line on the same grid for comparison will also make the curve more evident.
Perform an exponential regression on the data. Then estimate the amount of money the student placed in the account initially and the yearly interest rate.
f r (x) = 5995.64(1.0253)
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©val lawless/Shutterstock
Using a graphing calculator, the exponential regression formula is x
The student probably opened her account with $6000 and is earning 2.53% interest every year.
(
0.025 (Note: The formula used to get the data was f(x) = 6000 1 + ____ 12
)
12x
.
The data then had random numbers added to or subtracted from it. The original function can be simplified as follows:
(
0.025 f(x) = 6000 1 + ____ 12
)
12x
= 6000 (1.002083)
12x
= 6000 (1.002083 12)
x
= 6000 (1.0253)
x
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Have students investigate how many years it would take the student in the Lesson Performance Task to double the money in her account. Have students use the x equation y = 6000(1.0253) . Then ask students to estimate what interest rate would cause the money to double in just 18 years. Have them check their estimates and adjust them as needed. Students will find that the graphs of y = 6000(1.0253) and y = 12,000 intersect at about (27.8, 12,000), so the amount in the student’s account will have doubled after 27.8 years. They should find that an interest rate of 3.9% will cause the money to double in 18 years. x
19/04/14 11:44 AM
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Using Exponential Regression Models
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LESSON
15.4
Name
Comparing Linear and Exponential Models
Class
Date
15.4 Comparing Linear and Exponential Models Essential Question: How can you recognize when to use a linear model or an exponential model?
Common Core Math Standards
Resource Locker
The student is expected to: COMMON CORE
F-LE.A.1c
Suppose that you are offered a job that pays you $1000 the first month with a raise every month after that. You can choose a $100 raise or a 10% raise. Which option would you choose? What if the raise were 8%, 6%, or 4%?
Mathematical Practices COMMON CORE
MP.6 Precision
Language Objective
Find the monthly salaries for the first three months. Record the results in the table, rounded to the nearest dollar.
• For the $100 raise, enter 1000 into your graphing calculator, press ENTER, enter +100, press ENTER, and then press ENTER repeatedly. • For the 10% raise, enter 1000, press ENTER, enter x1.10, press ENTER, and then press ENTER repeatedly.
Describe the difference between a salary that changes by the same amount each year and a salary that changes by the same percent each year.
• For the other raises, multiply by 1.08, 1.06, or 1.04.
Monthly Salary after Indicated Monthly Raise
ENGAGE
A linear model should be used when the amount of increase or decrease in each successive interval is a constant. An exponential model is appropriate when the increase or decrease per successive interval grows.
$100
10%
8%
6%
4%
0
$1000
$1000
$1000
$1000
$1000
1
$1100
$1100
$1080
$1060
$1040
2
$1200 $1300
$1210 $1331
$1166 $1260
$1124 $1191
$1082 $1125
For each option, find how much the salary changes each month, both in dollars and as a percent of the previous month’s salary. Round the each percent to the nearest whole number. Record the values in the table.
Change in Salary per Month for Indicated Monthly Raise Interval
PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss why city officials might need to monitor population growth in their community. Then preview the Lesson Performance Task.
Month
3 © Houghton Mifflin Harcourt Publishing Company
Essential Question: How can you recognize when to use a linear model or an exponential model?
Comparing Constant Change and Constant Percent Change
Explore 1
Recognize situations in which a quantity grows or decays by a constant percent rate ... relative to another. Also F-LE.A.1a, F-LE.A.1b, F-LE.A.3
$100
10%
8%
6%
%
$
%
$
%
$
%
$
0–1
100
10
100
10
80
8
60
6
40
4
1–2
100
86
8
94
8
42 43
4
10
64 67
6
100
110 121
10
2–3
9 8
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be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
4
Date Class
r and aring Linea 15.4 Compnential Models Expo
Name
COMMON CORE
IN1_MNLESE389762_U6M15L4 749
an model or use a linear when to recognize t rate ... can you nt percen ion: How by a consta model? exponential grows or decays a quantity ns in which F-LE.A.3 t ize situatio F-LE.A.1b, Constan F-LE.A.1c Recogn r. Also F-LE.A.1a, nge and anothe relative to stant Cha
ing Con 1 Compar Explore Change Percent
HARDCOVER PAGES 749762
Resource Locker
You after that. every month 6%, or 4%? with a raise were 8%, first month if the raise $1000 the pays you choose? What a job that would you rounded you are offered Which option in the table, Suppose that raise or a 10% raise. the results $100 s. Record choose a three month
y g Compan
can
the first R, enter salaries for monthly press ENTE Find the calculator, t dollar. graphing to the neares into your R repeatedly. R, and then enter 1000 press ENTE $100 raise, and then press ENTE • For the R, enter x1.10, ENTER, press ENTE +100, press enter 1000, 10% raise, edly. • For the repeat R 1.06, or 1.04. press ENTE ly by 1.08, Raise raises, multip Monthly other ated 4% the Indic • For 6% Salary after 8% $1000 Monthly 10% $1000 $100 $1000 $1040 Month $1000 $1060 $1000 $1080 $1082 0 $1100 $1124 $1100 $1166 $1125 1 $1210 $1191 0 $1200 $126 1 2 $133 as a $1300 dollars and 3 , both in number. es each month the nearest whole t to salary chang much the each percen , find how . Round the option salary ’s For each us month the previo hly Raise 4% percent of ated Mont in the table. Indic values for h 6% Record the per Mont $ in Salary 8% % Change $ 40 10% % 6 $ 60 $100 % 42 8 $ 6 Interval 80 % 64 10 $ 43 8 100 6 86 10 67 10 100 8 110 0–1 94 9 10 100 121 1–2 8 100 2–3
Watch for the hardcover student edition page numbers for this lesson. % 4 4 4
© Houghto
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Harcour t
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C
Continue the calculations you did in Part A until you find the number of months it takes for each salary with a percent raise to exceed the salary with the $100 raise. Record the number of months in the table below.
EXPLORE 1
Number of Months until Salary with Percent Raise Exceeds Salary with $100 Raise 10%
8%
6%
4%
2
7
18
43
Comparing Constant Change and Constant Percent Change INTEGRATE TECHNOLOGY
Reflect
1.
Discussion Compare and contrast the salary changes per month for the raise options. Explain the source of any differences. A fixed raise increases by a constant amount monthly. A percent raise increases by an
Students can use graphing calculators to do repeated calculations for the Explore activity.
increasing amount monthly. A smaller percentage results in a slower increase. 2.
Discussion Would you choose a constant change per month or a percent increase per month? What would you consider when deciding? Explain your reasoning. Sample answer: A final decision should depend on which percent increase is offered and
AVOID COMMON ERRORS When calculating percent increases, remind students to take a percentage of the total from the previous interval, not from the original amount. For example, for a $1000 salary with a 10% raise each month, the increase in month 1 is 10% of $1000, but to find the increase in month 2, students need to take 10% of $1100, not 10% of the original $1000.
how long a person expects to keep the job.
Explore 2
Exploring How Linear and Exponential Functions Grow
Linear functions change by equal differences, while exponential functions change by equal factors. Now you will explore the proofs of these statements. x 2 - x1 and x 4 - x3 represent two intervals in the x-values of a function.
A
Complete the proof that linear functions grow by equal differences over equal intervals. x 2 - x1 = x 4 - x3
Given:
EXPLORE 2
ƒ is linear function of the form ƒ(x) = mx + b. ƒ(x 2) - ƒ(x 1) = ƒ(x 4) - ƒ(x 3)
Proof:
1. x 2 - x1 = x 4 - x3
Given
2. m(x 2 - x 1) = m x 4 - x3
Multiplication Property of Equality
3. mx 2 - mx 1 = mx 4 - mx 3
Distributive Property
4. mx 2 + b - mx 1 - b =
Addition & Subtraction Properties of Equality
mx 4 +
- mx 3 -
b
b
5. mx 2 + b - (mx 1 + b) =
( mx + b ) 6. f (x ) - f (x ) = ( f(x ) - f(x ) ) mx 4+ b 2
Module 15
Distributive property
© Houghton Mifflin Harcourt Publishing Company
Prove:
Exploring How Linear and Exponential Functions Grow INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Go over the goal of each proof. Students
3
1
4
3
Definition of ƒ(x)
750
should recognize that the first proof seeks to prove that in a linear function, the change in function values over equal intervals will be constant. The second proof shows that in an exponential function, the ratio of the function values over equal intervals will be constant.
Lesson 4
PROFESSIONAL DEVELOPMENT Learning Progressions
IN1_MNLESE389762_U6M15L4 750
In this lesson, students bring together their understanding of linear and exponential functions. They learn how to determine whether a given situation is best modeled by a linear or exponential function. A key concept is that if the dependent variable changes by equal differences over equal intervals a linear model is more appropriate, while if it changes by equal factors over equal intervals, an exponential model is more appropriate. Work with functions will continue when students learn about quadratic functions.
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QUESTIONING STRATEGIES In the proof for linear functions, why were b and –b added to both sides of the equation? What purpose did this step serve? The goal was to get each term of the equation, mx 1, mx 2, mx 3, and mx 4, in the form of mx + b. In that form, those quantities are equivalent to the functions f(x 1), f(x 2), f(x 4), and f(x 4), so they can be replaced by those functions in f(x) form.
Comparing Linear and Exponential Models
750
EXPLAIN 1
Complete the proof that exponential functions grow by equal factors over equal intervals. Given:
x 2 - x1 = x 4 - x3 g is an exponential function of the form g(x) = ab x.
Comparing Linear and Exponential Functions
Prove: Proof:
g(x 4) g(x 2) _ _ = g(x 3) g(x 1)
1. x 2 - x1 = x 4 - x3
QUESTIONING STRATEGIES
2. b (x
How do you know that you should write a linear equation to describe a salary that increases by $100 a month and an exponential equation to describe a salary that increases by 10% a month? An increase of $100 a month is an equal change over equal intervals, so it is a linear increase. An increase of the same percent each month is a change by equal factors over equal intervals, so it is an exponential function.
bx = _ bx 3. _ bx bx
2
- x1 )
2
Given
= b (x - x ) 4
If x = y, then b x = b y.
3
4
Quotient of Powers Property
1
3
ab x = _ ab x 4. _ ab x ab x
Multiplication Property of Equality
g (x 2 ) gx 4 5. _ = _ g(x 1) g(x 3)
Definition of g(x)
2
4
1
3
3.
In the previous proofs, what do x 2 - x 1 and x 4 - x 3 represent? x 2 - x 1 and x 4 - x 3 represent two intervals in the x-values of a function.
Explain 1
What does the intersection point of the graphs of the linear and exponential functions tell you? The intersection point tells how long it will take until both functions have the same value.
Comparing Linear and Exponential Functions
When comparing raises, a fixed dollar increase can be modeled by a linear function and a fixed percent increase can be modeled by an exponential function.
© Houghton Mifflin Harcourt Publishing Company
Example 1
Compare the two salary plans listed by using a graphing calculator. Will Job B ever have a higher monthly salary than Job A? If so, after how many months will this occur?
• Job A: $1000 for the first month with a $100 raise every month thereafter
• Job B: $1000 for the first month with a 1% raise every month thereafter Write the functions that represent the monthly salaries. Let t represent the number of elapsed months. Job B: S B(t) = 1000(1.01)
Job A: S A(t) = 1000 + 100t
t
Graph the functions on a calculator using Y 1 for Job A and Y 2 for Job B. Estimate the number of months it takes for the salaries to become equal using the intersect feature of the calculator. At x ≈ 364 months, the salaries are equal.
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Peer-to-Peer Activity Have students work with a partner. Provide them with the following problem: Job C provides a starting salary of $150 per day and a 0.5% increase each day. Job D provides a starting salary of $150 per day and a $1 increase each day. Will Job C ever have a higher daily salary than Job D? If so, after how many days will this occur? Have students predict the answers and then work out the problem independently. Students should compare their answers, discuss any differences, and discuss ways to make their predictions more accurate. On day 112, Job C will have a higher salary.
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Go to the estimated intersection point in the table feature. Find the first x-value at which Y 2 exceeds Y 1. Job B will have a higher monthly salary than Job A after 364 months.
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Ask students to explain how they know that
• Job A: $1000 for the first month with a $200 raise every month thereafter
the graphs for the linear and exponential function will eventually intersect. Students should recognize that an ever-increasing exponential function will eventually overtake a function that increases by a constant amount. Even with a small percent increase, sooner or later the total value of the exponential function will be sufficiently large to make the increase per unit time greater than the constant increase per unit time in the linear function. That said, there is no guarantee that the two quantities will be equal within a reasonable and practical amount of time.
• Job B: $1000 for the first month with a 4% raise every month thereafter Write the functions that represent the monthly salaries. Let t represent the number of elapsed months. Job A: S A(t) =
1000
+ 200 t
Job B: S B(t) =
1000
( 1.04 )
t
Graph the functions on a calculator and use this graph to estimate the number of months it takes for the salaries to become equal. At x ≈ 69 months, the salaries are equal. Job B will have a higher salary than Job A after 69 months. Reflect
4.
In Example 1A, which job offers a monthly salary that reflects a constant change, and which offers a monthly salary that reflects a constant percent change? Job A’s salary reflects a constant change. Job B’s salary represents a constant percent
change. 5.
Describe an exponential increase in terms of multiplication. An exponential increase can be obtained by repeatedly multiplying the same factor
greater than 1.
6.
© Houghton Mifflin Harcourt Publishing Company
Your Turn
• Job A: $2000 for the first month with a $300 raise every month thereafter • Job B: $1500 for the first month with a 5% raise every month thereafter
Job A: S A(t) = 2000 + 300t
Job B: S B(t) = 1500 + (1.05)
t
At x ≈ 50 months, the salaries are equal. Job B will have a higher salary than Job A after 50 months.
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Lesson 4
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Visual Cues
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It will be helpful for visual learners to study a graph that shows two functions representing job salaries, one starting at a low value and increasing exponentially, and the other starting at a higher value and increasing linearly. They can see that the exponential graph starts off below the linear graph; thus, the salary it represents is less. They can also see that at some point the exponential graph crosses the linear graph and goes above it, thus indicating that the salary represented by the exponential graph eventually will exceed the salary represented by the linear graph.
Comparing Linear and Exponential Models
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Choosing between Linear and Exponential Models
Explain 2
EXPLAIN 2
Both linear equations and exponential equations and their graphs can model real-world situations. Determine whether the dependent variable appears to change by a common difference or a common ratio to select the correct model.
Choosing Between Linear and Exponential Models
Example 2
QUESTIONING STRATEGIES
How do you decide whether to use a linear or exponential regression model for a data set? First calculate the difference in y-values over each equal interval and the ratio of y-values over each equal interval. If the differences are roughly equal, use a linear model. If the ratios are roughly equal, use an exponential model.
Determine whether each situation is better described by an increasing or decreasing function, and whether a linear or exponential regression should be used. Then find a regression equation for each situation by using a graphing calculator.
A gas has an initial pressure of 165 torr. Its pressure was then measured every 5 seconds for 25 seconds.
Pressure over Time
Change per Interval Factor
P (t ) _ n
Time (s)
Pressure (torr)
0
165
5
153
−12
0.93
10
142
−11
0.93
15
129
−13
0.91
20
116
−13
0.90
25
102
−14
0.88
Difference P(t n) - P(t n - 1)
P(t n - 1)
The dependent variable is pressure, and it is decreasing while the number of seconds is increasing. This means that the function is decreasing. Note that because the factor changes are relatively close to equal while the difference changes are not, an exponential regression model should be used.
© Houghton Mifflin Harcourt Publishing Company
Perform the exponential regression analysis and evaluate the fit.
Note that the r-value suggests a good fit.
The analysis of residuals suggests a good fit. Module 15
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Lesson 4
LANGUAGE SUPPORT IN1_MNLESE389762_U6M15L4 753
Connect Context Discuss the various ways that a change per unit of time may be expressed in verbal descriptions. For example, the phrases yearly, each year, and annually are all ways of saying per year. In statements about changing salaries such as “employees receive a $0.75 per hour raise each year,” remind students to read carefully to distinguish the rate of pay from the rate at which the pay is changing.
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A closed container holding an unspecified gas has an initial pressure of 44 torr. The container was then placed over a flame and its pressure was measured every 5 minutes for 25 minutes.
Pressure over Time
AVOID COMMON ERRORS When given a table of data values, students may automatically calculate differences and ratios of y-values over the given intervals. Remind them that the change in y must be evaluated over equal intervals. They should first check whether the difference in x-values is the same for each interval. If it is not, they will need to adjust their calculations.
Change per Interval Factor
Time (min)
Pressure (torr)
0
44
5
49
10
60
15
72
20
90
25
105
Difference P(t n) - P(t n - 1) 5 11 12 18 15
P (t ) _ n
P(t n - 1) 1.11 1.22 1.20 1.25 1.17
Is the function increasing or decreasing? Explain. Pressure, which is the dependent variable, is
increasing as time increases. This means that the function is increasing. Which changes are closer to being equal, the differences or the factors? factors Which type of regressions should be used? exponential Perform the regression analysis and evaluate the fit.
good fit. The analysis of residuals suggests a
good fit.
Reflect
7.
What would the residual plot look like if an exponential regression was not a good fit for a function? If an exponential function was not a good fit for a function, the residual plot would contain
points very far away from the origin.
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© Houghton Mifflin Harcourt Publishing Company
Note that the r-value suggests a
Lesson 4
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Comparing Linear and Exponential Models
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Your Turn
ELABORATE
Determine whether this situation is better described by an increasing or decreasing function, and whether a linear or exponential regression should be used. Then find a regression equation.
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 As students work with functions modeling
8.
Pressure over Time Pressure (torr)
0
432
5
454
10
499
15
534
20
582
25
611
Difference P(t n) - P(t n - 1)
P (t ) _ n
P(t n - 1) 1.05 1.10 1.07 1.09 1.05
22 45 35 48 29 x
Since the r-value is close to 1 and the residuals are close together, this is a good fit.
Elaborate
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Greg Lawler/Alamy
9.
In the long term, which type of raise will guarantee a larger paycheck: a fixed raise or a percentage raise? In the long term, a percentage raise will always guarantee a larger paycheck.
10. What type of function is typically represented by a linear function? A function with a constant increase or decrease is typically represented by a linear
function. 11. Essential Question Check-In An exponential growth model is appropriate when the increase per successive interval grows
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Lesson 15.4
Time (min)
Exponential regression: y ≈ 429.1 (1.01)
SUMMARIZE THE LESSON
755
Change per Interval Factor
real-world examples of increase and decrease, review what the parameters of the equations mean in this context. In a linear equation of the form y = mx + b, the coefficient m is the rate of change, and the constant b is the initial value, or the value of y when x is 0. In an exponential equation of the form y = ab x, the coefficient a is the initial value, and the base b is the growth factor.
How can you recognize when to use a linear model or an exponential model? A linear model should be used when the amount of increase or decrease in each successive interval is a constant. An exponential model is appropriate when the increase or decrease per successive interval grows, or changes.
A barrel of gasoline has an initial pressure of 432 torr. Its pressure was then measured every 5 minutes for 25 minutes.
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Evaluate: Homework and Practice
EVALUATE • Online Homework • Hints and Help • Extra Practice
State whether each situation is best represented by an exponential or linear function. Then write an exponential or linear function for the model and state whether the model is increasing or decreasing. 1.
Enrollment at a school is initially 454 students and grows by 3% per year.
exponential; E(t) = 454(1.03) ; increasing t
2.
A salesperson initially earns $50,434 dollars per year and receives a yearly raise of $675.
ASSIGNMENT GUIDE
linear; S(t) = 50, 434 + 675t; increasing
3.
A customer borrows $450 at 5% interest compounded annually.
exponential; C(t) = 450(1.05) ; increasing t
4.
A wildlife park has 35 zebras and sends 1 zebra to another wildlife park each year.
5.
The value of a house is $546,768 and decreases by 3% each year. t exponential; H(t) = 546, 768(.97) ; decreasing
6.
The population of a town is 66,666 people and decreases by 160 people each year.
7.
A business has a total income of $236,000 and revenues go up by 6.4% per year. t exponential; l(t) = 236, 000(1.064) ; increasing
linear; Z(t) = 35 - t; decreasing
linear; P(t) = 66, 666 - 160t; decreasing
Use a graphing calculator to answer each question. 8.
Finance Employees A and B each initially earn $18.00 per hour. If Employee A receives a $1.50 per hour raise each year and Employee B receives a 4% raise each year, when will Employee B make more per hour than Employee A? Employee B will make more than Employee A after about 35 years.
10. Finance Account A and B each start out with $400. If Account A earns $45 each year and Account B earns 5% of its value each year, when will Account B have more money than Account A?
Account B will have more money than Account A after about 31 years.
Stock B will be worth more than Stock A after about 80 months.
Exercise
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Exercise 21
Explore 2 Exploring How Linear and Exponential Functions Grow
Exercises 1–7, 20
Example 1 Comparing Linear and Exponential Functions
Exercises 8–15, 22–23
Example 2 Choosing Between Linear and Exponential Models
Exercises 16–19, 24
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Depth of Knowledge (D.O.K.)
Explore 1 Comparing Constant Change and Constant Percent Change
appropriate viewing window for the graphs they generate to solve problems. They can either manually input minimum and maximum x- and y-values appropriate to the range of the data, or they can use the various options on the calculator’s ZOOM menu. They may need to adjust the window a few times until they can see the intersection point of the graphs.
11. Finance Stock A starts out with $900 and gains $50 each month. Stock B starts out with $800 and gains 11% each month. When will Stock B be worth more money than Stock A?
Module 15
Practice
INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Remind students how to find an
© Houghton Mifflin Harcourt Publishing Company
9.
Statistics Companies A and B each have 100 employees. If Company A increases its workforce by 31 employees each month and Company B increases its workforce by an average of 10% each month, when will Company B have more employees than Company A? Company B will have more employees than Company A after about 21 months.
Concept and Skills
COMMON CORE
Mathematical Practices
1–7
1 Recall of Information
MP.2 Reasoning
8–15
2 Skills/Concepts
MP.4 Modeling
16–19
2 Skills/Concepts
MP.6 Precision
20
1 Recall of Information
MP.2 Reasoning
21
2 Skills/Concepts
MP.4 Modeling
22
2 Skills/Concepts
MP.4 Modeling
23
2 Skills/Concepts
MP.2 Reasoning
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12. Finance Accounts A and B both start out with $800. If Account A earns $110 per year and Account B earns 3% of its value each year, when will Account B have more money than Account A?
MODELING When graphs have enormous scales, they may seem abstract. Students may fail to grasp the idea that the intersection of two graphs identifies where they are equal in value. To help students see this, have them graph y = 2x + 2 and y = 2 x by hand. Students can readily see that both functions have the same value at x = 3.
Account B will have more money than Account A after about 87 years.
13. Finance Two factory workers, A and B, each earn $24.00 per hour. If Employee A receives a $0.75 per hour raise each year and Employee B receives 1.9% raise each year, when will Employee B make more per hour than Employee A?
Employee B will make more than Employee A after about 50 years. 14. Statistics Two car manufacturers, A and B, each have 500 employees. If Manufacturer A increases its workforce by 15 employees each month and Manufacturer B increases its workforce by 1% each month, when will Manufacturer B have more employees?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Vasily Smirnov/Shutterstock
Manufacturer B will have a larger workforce after about 192 months.
15. Finance Stock A is initially worth $1300 and loses $80 each month. Stock B is initially worth $400 and gains 9.5% each month. When will Stock B be worth more than Stock A?
Stock B will be worth more than Stock A after about 7 months.
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Exercise 24
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Lesson 15.4
Lesson 4
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Depth of Knowledge (D.O.K.) 3 Strategic Thinking
COMMON CORE
Mathematical Practices
MP.3 Logic
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Physics Each table illustrates an unknown container’s pressure over time. Determine whether the table is best described by an increasing or decreasing function and whether a linear or exponential regression should be used. Then find a regression equation for each situation. 16.
difference y2 - y1
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Before students begin manipulating data from
2 factor __ y1
y
x
y
1
49
2
58
9
1.18
3
70
12
1.21
4
83
13
1.19
5
101
18
1.22
a table, have them observe the pattern of the raw data. Discuss how the difference between successive values of the function changes from one interval to the next. Students should recognize that if the differences are the same, a linear model may fit the data. If differences increase or decrease in successive intervals, an exponential model may fit the data. Then have students explain what the change means in terms of the real-world situation.
Since the values of y are increasing over time, the function is increasing. Since the factors of y are close together while the differences are relatively far apart, an exponential regression should be used. y ≈ 40.7 (1.20) Because the residuals are close to 0, the equation is a good fit for the function described by the table. x
17.
difference y2 - y1
2 factor __ y1
y
x
y
1
31
2
32
1
1.03
3
34
2
1.06
4
35
1
1.03
5
37
2
1.06 © Houghton Mifflin Harcourt Publishing Company
Since the values of y are increasing over time, the function is increasing. Since the factors of y are close together while the differences are relatively close, a linear regression should be used. y ≈ 1.5 (29.3) Because the residuals are close to 0, the equation is a good fit for the function described by the table. x
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Comparing Linear and Exponential Models
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18.
AVOID COMMON ERRORS
x
After using a calculator to perform a regression, some students may write the function incorrectly. Remind them that, for a linear regression, the parameter a represents the slope and b represents the y-intercept, so the function is y = ax + b. For an exponential regression, a is the coefficient and b is the base of the exponent, so the function is y = ab x.
y
difference y2 - y1
2 factor __ y1
y
1
46
2
61
3
83
4
107
24
1.29
5
143
36
1.34
15
1.33 1.36
Since the values of y are increasing over time, the function is increasing. Since the factors of y are close together while the differences are relatively far apart, an exponential regression should be used.
y ≈ 34.8 (1.33) Because the residuals are close to 0, the equation is a good fit for the function described by the table. x
19.
difference y2 - y1
2 factor __ y1
35
13
1.59
60
25
1.71
4
104
44
1.73
5
189
85
1.82
x
y
1
22
2 3
y
© Houghton Mifflin Harcourt Publishing Company
Since the values of y are increasing over time, the function is increasing. Since the factors of y are not close together while the differences are relatively far apart, no regression should be used.
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20. Using the given exponential functions, state a and b. a. y = 3(4)
b. y = -5(8) a = -5, b = 8 x
x
a = 3, b = 4
d. y = -5(0.9) a = -5, b = 0.9 x
c.
y = 4(0.6)
MULTIPLE REPRESENTATIONS
x
Students may benefit from using simple numbers and number sense to justify algebraic steps in proofs. For example, to justify
a = 4, b = 0.6
e. y = 2 x a = 1, b = 2
x2 – x1 = x4 – x3
21. Suppose that you are offered a job that pays you $2000 the first month with a raise every month after that. You can choose a $400 raise or a 15% raise. Which option would you choose? What if the raise were 10%, 8%, or 5%?
m(x 2 – x 1) = m(x 4 – x 3)
Monthly Salary after Indicated Monthly Raise Month
$400
15%
10%
8%
5%
0
$2000
$2000
$2000
$2000
$2000
1
$2400
$2300
$2200
$2160
$2100
$2205 $2315
2
$2800
$2645
$2420
$2333
3
$3200
$3042
$2662
$2519
mx 2 – mx 1 = mx 4 – mx 3 write 5–2=7–4 3(5 – 2) = 3(7 – 4) 3(5) – 3(2) = 3(7) – 3(4) 15 – 6 = 21 – 12
Change in Salary per Month for Indicated Monthly Raise Interval
$400
15%
10%
8%
5%
$
%
$
%
$
%
$
%
$
%
0–1
400
20
300
15
200
10
160
8
100
5
1–2
400
17
345
15
220
10
173
8
105
5
2–3
400
14
397
15
242
10
186
8
110
5
9=9 Students can readily see that the algebraic steps are “legal” because the original equation, which started out as true, remained true even though it was manipulated.
15%
10%
8%
5%
5
15
22
49
© Houghton Mifflin Harcourt Publishing Company
Number of Months until Salary with Percent Raise Exceeds Salary with $400 Raise
The salary chosen would depend on the length of the job and the percentage change in salary offered.
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Comparing Linear and Exponential Models
760
JOURNAL
H.O.T. Focus on Higher Order Thinking
22. Draw Conclusions Liam would like to put $6000 in savings for a 5-year period. Should he choose a simple interest account that pays an interest rate of 5% of the principal (initial amount) each year or a compounded interest account that pays an interest rate of 1.5% of the total account value each month?
Ask students to draw a table in their journals comparing a linear function with a positive slope to an exponential function with a positive coefficient. Linear Function Change per equal constant interval Ratio over equal intervals f(x n) ______ f(x n-1) Equation
Monthly:
Yearly:
y = 6000(1.05)
Exponential Function
= 6000(1.05)
y = 6000(1.015)
x
x
= 6000(1.015)
5
60
= 6000(1.276)
= 6000(2.44)
= $7657.69
increasing
= $14, 659.32
$14, 659.32 - $7657.69 = $7001.63 Liam should choose a monthly account as it will earn roughly $7001.63 more dollars in a 5-year period.
decreasing
constant 23. Critical Thinking Why will an exponential growth function always eventually exceed a linear growth function?
f(x) = a + bx
Since the exponential growth function eventually curves upward while a linear growth function continues in a straight line with a positive slope, the exponential growth function will eventually exceed the linear growth function.
f(x) = ab x
© Houghton Mifflin Harcourt Publishing Company
24. Explain the Error JoAnn analyzed the following data showing the number of cells in a bacteria culture over time.
Time (min)
0
6.9
10.8
13.5
15.7
17.4
Cells
8
16
24
32
40
48
She concluded that since the number of cells showed a constant change and the time did not, neither a linear function nor an exponential function modeled the number of cells over time well. Was she correct?
JoAnn was incorrect. Because the time intervals are not uniform, she cannot make a determination by just looking at the data. An exponential function is a good model as exponential regression gives an equation with r = 0.9999.
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Lesson Performance Task
AVOID COMMON ERRORS Some students may use the growth factor 1.05 for City B because they incorrectly converted 0.5% to the decimal 0.05 instead of 0.005. Remind students that percent means per hundred and hundredths have two decimal places. Therefore, when converting a percent to a decimal, move the decimal point two places to the left.
Two major cities each have a population of 25,000 people. The population of City A increases by about 150 people per year. The population of City B increases by about 0.5% per year.
a. Find the population increase for each city for the first 5 years. Round to the nearest whole number, if necessary. Then compare the changes in the populations of each city per year. b. Will City B ever have a larger population than City A? If so, what year will this occur?
a. First, find the yearly population of each city for the first 5 years.
QUESTIONING STRATEGIES
Next, find the population increase for each city for the first 5 years.
Yearly Population
What type of function describes each city’s growth? Explain. The growth of City A is described by a linear function, because it increases by a constant amount per year. The growth of City B is described by an exponential function because it increases by a constant percent per year.
Yearly Population Increase
City A
City B
Year
City A
City B
0
25,000
25,000
0–1
150
125
1
25,150
25,125
1–2
150
126
2
25,300
25,251
2–3
150
126
3
25,450
25,377
3–4
150
127
4
25,600
25,504
4–5
150
128
5
25,750
25,632
The population of City A is increasing at a constant 150 people per year. The population of City B is increasing at small intervals and not at a constant rate.
b. Let t represent the number of elapsed years. City A: P A(t) = 25, 000 + 150t
City B: P B(t) = 25, 000x + 1.005 t The estimated intersection point is (72, 35,794). City B will have a larger population after about 72 years.
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How do you use the initial population value in the equation for each function? In the linear function for City A, the initial population is the y-intercept. In the exponential function for City B, the initial population is the coefficient.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Greg Henry/Shutterstock
Year
Lesson 4
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Have students research factors that affect the growth rate of any city. Then have students tell which factors may cause a decrease in population and which may cause an increase. Students should find that births, deaths, immigration, and emigration affect growth rate. Deaths and emigration cause a decrease in population, while births and immigration cause an increase in population. The growth rate is determined using this equation: growth rate = birth rate – death rate + immigration rate – emigration rate
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Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Comparing Linear and Exponential Models
762
MODULE
15
MODULE
STUDY GUIDE REVIEW
15
Exponential Equations and Models
Study Guide Review
Essential Question: How can you use exponential equations to represent real-world situations?
ASSESSMENT AND INTERVENTION
KEY EXAMPLE
Key Vocabulary
exponential decay (decremento exponencial) exponential growth (crecimiento exponencial) exponential regression (regresión exponencial)
(Lesson 15.2)
A comic book is sold for $3, and its value increases by 6% each year after it is sold. Write an exponential growth function to find the value of the comic book in 25 years. Then graph it and state its domain and range. What does the y-intercept represent? Write the exponential growth function for this situation.
Assign or customize module reviews.
y = a(1 + r)
t
= 3(1 + 0.06)
= 3(1.06)
MODULE PERFORMANCE TASK
t
t
Find the value in 25 years. y = 3(1.06)
t
= 3(1.06)
25
≈ 12.88
COMMON CORE
After 25 years, the comic book will be worth approximately $12.88.
Mathematical Practices: MP.1, MP.2, MP.4, MP.5, MP.6 A-CED.A.2, F-BF.A.1, F-LE.A.1c, F-LE.A.2
Create a table of values to graph the function.
• What is a half-life? Nuclear isotopes decay into stable molecules at a predictable rate. A half-life is the amount of time it takes for half of an isotope to decay. • How can you calculate nuclear decay? You can x 1 __ use the formula y = a 2 , where a is the original amount of the substance, x is the number of half-lives, and y is the remaining amount of the substance.
()
• Should I use 365 days per year or 365.25 days per year? The difference is negligible in this case, so either can be used.
(t, y)
t
y
0
3
(0, 3)
5
4.01
(5, 4.01)
10
5.37
(10, 5.37)
15
7.19
(15, 7.19)
20
9.62
(20, 9.62)
25
12.88
(25,12.88)
30
17.23
(30, 17.23)
y 16 Cost ($)
Students should begin this problem by figuring out what information they need before they find the amount of iodine-131 released. Here are some questions they might ask.
© Houghton Mifflin Harcourt Publishing Company
SUPPORTING STUDENT REASONING
12 8 4 t 0
10
20
Time (years)
The domain is the set of real numbers t such that t ≥ 0. The range is the set of real numbers y such that y ≥ 3.
The y-intercept is the value of y when t = 0, which is the time when the comic book was sold.
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Study Guide Review
SCAFFOLDING SUPPORT
IN1_MNLESE389762_U6M15MC 763
• Students who assume the decay for cesium-137 over 40 days is negligible and solve for a mass of 100 g of iodine-131 will get an initial mass of 3200 g for cesium-137. This is a difference of less than 0.3% and is acceptable. • Watch for students who neglect to convert years to units of days.
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EXERCISES Solve each equation for x. (Lesson 15.1) 1.
SAMPLE SOLUTION
3(2) = 96 x
2.
5 x = 25 ___ 25
x=5 3.
First, find the amount of cesium-137 that remains after 40 days. The proportion that relates 40 days to 40 30 years is ________ , so 40 days is about 0.00365 of a 365.25 · 30 half-life. Substitute and solve:
x=4
The value of a textbook is $120 and decreases at a rate of 12% per year. Write a function to model the situation, and then find the value of the textbook after 9 years. (Lesson 15.2)
(__12 )
f(t) = 120(0.88) ; $37.98 t
y = 100
Find an exponential model for the data in the given table. (Lesson 15.3) 4.
x
0
1
2
3
4
5
6
7
8
9
10
y
9
12.85
16.89
28.15
42.58
65.1
99.34
153
237.6
339.2
478.61
f(x) = 8.37(1.51)
40 For iodine-131, 40 days is equivalent to ___ =5 8 half-lives.
(__12 ) 1 99.75 = a(___ 32 )
State whether each situation is best represented by an exponential or linear function. Then write an exponential or linear function for the model and state whether the model is increasing or decreasing. (Lesson 15.4) A customer borrows $950 at 6% interest compounded annually. t exponential; C(t) = 950(1.06) ; increasing
6.
The population of a town is 8548 people and decreases by 90 people each year. linear; P(t) = 8548 - 90t ; decreasing
≈ 99.75
After 40 days, there are about 99.75 g of cesium-137 and the same amount of iodine-131.
x
5.
0.00365
99.75 = a
5
a = 3192
Substitute. Simplify. Solve.
There were 3192 g of iodine-131 released in the accident.
MODULE PERFORMANCE TASK
Half-Life The half-life of iodine-131 is 8 days, and the half-life of cesium-137 is 30 years. Both of these isotopes can be released into the environment during a nuclear accident. © Houghton Mifflin Harcourt Publishing Company
Suppose that a nuclear reactor accident released 100 grams of cesium-137 and an unknown amount of iodine-131. After 40 days the amount of iodine-131 is equal to the amount of cesium-137. About how much iodine-131 was released by the accident? Start by listing in the space below how you plan to tackle the problem. Then use your own paper to complete the task. Be sure to write down all your data and assumptions. Then use numbers, graphs, tables, or algebra to explain how you reached your conclusion.
Module 15
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Study Guide Review
DISCUSSION OPPORTUNITIES
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• Ask students why knowing the initial mass of radioisotopes released can be useful. • Have students share techniques and strategies. Discuss and evaluate different approaches that classmates used to solve the problem. Assessment Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem.
Study Guide Review 764
Ready to Go On?
Ready to Go On?
ASSESS MASTERY
15.1–15.4 Exponential Equations and Models
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
• Online Homework • Hints and Help • Extra Practice
Mike has a savings account with the bank. The bank pays him annual interest of 1.5%. He has $4000 and wonders how much he will have in the account in 5 years. Write an exponential function to model the situation and then find how much he will have. (Lesson 15.1)
1.
S(t) = 4000(1.015) ; about $4309.14 t
ASSESSMENT AND INTERVENTION
State each function’s domain, range, and end behavior. (Lesson 15.2) ƒ(x) = 900(0.65) ⎧ ⎫ Domain: ⎨x| - ∞ < x < ∞⎬ ⎩ ⎭ ⎧ ⎫ Range: ⎨y | y > 0⎬ ⎩ ⎭ End behavior: As x → -∞, y → ∞ x
2.
and as x → ∞, y → 0.
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Time, (x) Temperature, (y)
Differentiated Instruction Resources • Reading Strategies • Success for English Learners • Challenge Worksheets Assessment Resources
© Houghton Mifflin Harcourt Publishing Company
• Reteach Worksheets
and as x → ∞, y → ∞.
0
4
8
12
16
450
340
240
190
145
5.
after about 12 years
ESSENTIAL QUESTION How can you identify an exponential equation?
6.
Possible Answer: An exponential equation has the form f(x) = ab x , for real numbers a, b, and x where a ≠ 0, b > 0, and b ≠ 1. In an exponential equation, as the input values increase by 1, the successive output values are related by a constant ratio.
Module 15
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Module 15
x
Create a model describing the data and use it to predict the temperature after 20 minutes. (Lesson 15.3) x f(x) = 445(0.931) ; 106.5 degrees Account A and B each start out with $600. If Account A earns $50 each year and Account B earns 6% of its value each year, after how many years will Account B have more money than Account A? (Lesson 15.4)
• Leveled Module Quizzes
765
ƒ(x) = 400(1.23) ⎧ ⎫ Domain: ⎨x| -∞ < x < ∞⎬ ⎩ ⎭ ⎧ ⎫ Range: ⎨y | y > 0⎬ ⎩ ⎭ End behavior: As x → -∞, y → 0
The table shows the temperature of a pizza over three-minute intervals after it is removed from the oven.
4.
ADDITIONAL RESOURCES Response to Intervention Resources
3.
Study Guide Review
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Common Core Standards
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Content Standards
Mathematical Practices
1
A-CED.A.1, A-SSE.B.3, F-BF.A.1, F-LE.A.2, S-ID.B.6, F-LE.A.1
MP.4
14.2
2
F-IF.C.7, F-IF.B.5
MP.2
14.2
3
F-IF.C.7, F-IF.B.5
MP.2
14.3
4
A-SSE.B.3, F-BF.A.1, F-LE.A.2, F-LE.A.1, S-ID.B.6
MP.4
14.4
5
A-SSE.B.3, F-LE.A.2, F-LE.A.1, S-ID.B.6
MP.4
Lesson
Items
14.1
MODULE MODULE 15 MIXED REVIEW
MIXED REVIEW
Assessment Readiness
Assessment Readiness
1. Consider the end behavior of f(x) = 75(1.25) . Select True or False for each statement. A. As x → -∞, y → -∞. True False B. As x → -∞, y → 0. True False x
C. As x → ∞, y → ∞.
15
True
ASSESSMENT AND INTERVENTION
False
2. An engineer took the following measurements: 3.22 cm, 14.1 cm, 18 cm, and 24.025 cm. Choose True or False for each statement. A. The most precise measurement has 4 significant digits.
True
False
B. Written using the correct number of significant digits, the sum of the measurements should be rounded to the ones place.
True
False
C. The least precise measurement has 2 significant digits.
True
False
Assign ready-made or customized practice tests to prepare students for high-stakes tests.
ADDITIONAL RESOURCES
3. Solve 36(3) = 4. What is the value of x? Explain how you got your answer. x
Assessment Resources
-2; I divided both sides by 36 and got 3 x = _91 . Then I rewrote __19 as a power of 3 to get 3 x = 3 -2. Applying the equality of bases property, I found that x equals -2.
• Leveled Module Quizzes: Modified, B
4. Consider the following situation: enrollment at a school is initially 322 students and grows by 4% per year. Write an equation to represent this situation, and use it to predict the number of students at the school in 5 years.
AVOID COMMON ERRORS
E(t) = 322(1.04) ; In 5 years, the school will have approximately 392 students. t
© Houghton Mifflin Harcourt Publishing Company
Item 1 Some students may have difficulty picturing the end behavior of an exponential function as it approaches negative infinity. Encourage them to substitute large negative numbers into the function to help them find the end behavior.
Module 15
COMMON CORE
Study Guide Review
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Common Core Standards
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Content Standards Mathematical Practices
Lesson
Items
14.2
1
F-IF.B.4
MP.1
1.3
2*
N-Q.A.3
MP.6
14.1
3
N-RN.A.2
MP.2
14.2
4
F-BF.A.1
MP.4
* Item integrates mixed review concepts from previous modules or a previous course.
Study Guide Review 766
UNIT
6
UNIT 6 MIXED REVIEW
Assessment Readiness
MIXED REVIEW
Assessment Readiness
1. Is the given equation equivalent to V = IR?
ASSESSMENT AND INTERVENTION
A. R = IV
Yes
No
V B. R = _ I V C. I = _ R
Yes
No
Yes
No
• Online Homework • Hints and Help • Extra Practice
2. Consider the graph of 16x − 2y = 48. Select True or False for each statement. A. The slope is 8. True False
Assign ready-made or customized practice tests to prepare students for high-stakes tests.
True
False
C. The x-intercept is 3.
True
False
3. Consider the sequence 8, 4, 0, -4…. Determine if each statement is True or False. A. It is a geometric sequence. True False
ADDITIONAL RESOURCES Assessment Resources • Leveled Unit Tests: Modified, A, B, C • Performance Assessment
B. The fifth term is -8.
True
False
C. f(10) = -28.
True
False
4. Write an explicit and recursive rule for the geometric sequence -5, 10, -20, 40… and use it to find the 12 th term of the sequence. Is each statement correct? © Houghton Mifflin Harcourt Publishing Company
AVOID COMMON ERRORS Item 1 Some students may not know how to approach the problem. Suggest that they start by solving each expression for V. Then, the comparison is straightforward.
B. The y-intercept is 4.
A. The recursive rule is f(1) = -2; f(n) = 5 · f(n - 1).
Yes
No
B. The explicit rule is n-1 f(n) = -5(-2) .
Yes
No
C. The 12 th term is -20, 480.
Yes
No
(2)
A. As x → -∞, y → -∞
True
False
B. As x → ∞, y → 0
True
False
C. As x → ∞, y → -∞
True
False
Unit 6
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Common Core Standards
Items
767
Unit 6
x
1 5. Consider the end behavior of f(x) = -6 _ . Is each statement True or False?
Content Standards Mathematical Practices
1*
A-CED.A.4
MP.2
2*
F-IF.A.2, A-CED.A.2
MP.7
3*
F-LE.A.2
MP.1
4
F-BF.A.1
MP.4
5
F-IF.C.7
MP.4
6*
A-REI.B.3
MP.2
* Item integrates mixed review concepts from previous modules or a previous course.
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6. Solve each equation. Is the given solution correct? 1 A. 5(2x - 7) = -6x - 27; x = _ 2 2 x = -2x - 2; x = -3 B. 6 - _ 3 C. 9p = 3(4 - p) + 12; p = 12
Yes
No
PERFORMANCE TASKS
Yes
No
There are three different levels of performance tasks:
Yes
No
* Novice: These are short word problems that require students to apply the math they have learned in straightforward, real-world situations.
−x
7. Describe the end behavior of y = −2(3) . As x approaches infinity, y approaches 0. As x approaches negative infinity, y approaches
** Apprentice: These are more involved problems that guide students step-by-step through more complex tasks. These exercises include more complicated reasoning, writing, and open ended elements.
negative infinity.
()
X
1 . What are the domain and range of the function? 8. Graph f(x) = 4 __ 2 16
y
8
***Expert: These are open-ended, nonroutine problems that, instead of stepping the students through, ask them to choose their own methods for solving and justify their answers and reasoning.
x -4
-2
0
2
4
-8
-16
Domain: {-x ⎪ = -∞ < x < ∞}; Range: {y ⎪ y > 0}
SCORING GUIDES
9. Solve 3(16) 4 = 192 for x. Show your work. x _
Item 10 (2 points) Award the student 1 point for correctly finding the first four terms, and 1 point for correctly explaining why it is not a geometric sequence.
_x 4
3(16) = 192 _x
(16)4 = 64 x 4(__ 4) 6 =2
2
x=6
Performance Tasks 10. Francesca invents the recursive rule f(1) = 3, f(n) = f(n − 1) • f(n − 1). Write the first four terms of the sequence. Is the sequence geometric? Explain why or why not. 3, 9, 81, 6561; The sequence is not geometric, because successive terms are not related by a common ratio.
Unit 6
© Houghton Mifflin Harcourt Publishing Company
2 = 26 x
768
COMMON CORE IN1_MNLESE389762_U6UC 768
Common Core Standards
Items
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Content Standards Mathematical Practices
7*
F-IF.A.1
MP.6
8
F-IF.C.7
MP.4
9
A-CED.A.1
MP.2
* Item integrates mixed review concepts from previous modules or a previous course.
Unit 6
768
11. Billy earns money by mowing lawns for the summer. He offers two payment plans. Plan 1: Pay $250 for the entire summer. Plan 2: Pay $1 the first week, $2 the second week, $4 the third week, and so on.
SCORING GUIDES Item 11 (6 points) a. 1 point for correct identification of geometric sequence 2 points for explanation
A. Do the payments for Plan 2 form a geometric sequence? Explain. B. If you were one of Billy’s customers, which plan would you choose? (Assume that the summer is 10 weeks long.) Explain your choice. A. Yes, it is a geometric sequence with r = 2.
b. 1 point for correctly choosing plan 1 2 points for explanation
B. Possible answer: Plan 1; under Plan 2, the cost for the 10 th week alone is $512, which is more than the cost for the entire summer under Plan 1.
Item 12 (6 points) a. 1 point for correct amount b. 1 point for correct answer 12. As a promotion, a clothing store draws the name of one of its customers each week. The prize is a coupon for the store. If the winner is not present at the drawing, he or she cannot claim the prize, and the amount of the coupon x increases for the following week’s drawing. The function f (x) = 20(1.2) gives the amount of the coupon in dollars after x weeks of the prize going unclaimed.
c. 1 point for correct amount d. 1 point for correct percent e. 2 points for answer and explanation
A. What is the amount of the coupon after 2 weeks of the prize going unclaimed? B. After how many weeks of the prize going unclaimed will the amount of the coupon be greater than $100?
© Houghton Mifflin Harcourt Publishing Company
C. What is the original amount of the coupon? D. Find the percent increase each week. E. Do you think it would be wise for the owner of the store to set a limit on the number of weeks a prize can go unclaimed? Why or why not? A. 28.8 B. 9 whole weeks C. 20 D. 20% E. Yes; If the prize is claimed then the total cost through several weeks grow linearly. When the prize is not claimed it grows exponentially. After several weeks the exponential curve is more expensive.
Unit 6
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math in careers
b. For Stock B, which model fits the graph, exponential growth or exponential decay? Find the initial value and the growth or decay factor.
Financial Research Analyst In this Unit Performance Task, students can see how a financial research analyst uses mathematics on the job.
16 Value ($)
a. For Stock A, which model fits the graph, exponential growth or exponential decay? Find the initial value and the growth or decay factor.
MATH IN CAREERS
y
Financial research analyst The graph shows the value of two different shares of stock over the period of four years since they were purchased. The values have been changing exponentially.
12
Stock A
8 Stock B
4
For more information about careers in mathematics as well as various mathematics appreciation topics, visit the American Mathematical Society http://www.ams.org
t 0
c. According to the graph, after how many years was the value of Stock A about equal to the value of Stock B? What was that value?
1
2
3
Time (years)
d. After how many years was the value of Stock A about twice the value of Stock B? Explain how you found your answer.
SCORING GUIDES
a. exponential decay; initial value: $16, decay factor: 0.75
Task (6 points)
b. exponential growth; initial value : $5, growth factor: 1.2 c. About 2.5 years; in 2.5 years, the value of both stocks was about $8 per share.
a. 1 point for correct model and initial value 1 point for correct decay value
d. About 1 year; in 1 year, the value of Stock B was about $5 and the value of Stock A was about $12, which was twice the value of Stock B.
b. 1 point for correct model and initial value 1 point for correct decay value c. 1 point for correct year and value d. 1 point for correct year and explanation
© Houghton Mifflin Harcourt Publishing Company
Unit 6
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Unit 6
770
