Algebra Review 2 – Quadratic Functions Introduction 2.1 – Warmup 1 - Quadratic Function Standard Form
Name ______________________ Per ____ Date _______________
Motivation Tommy the turtle stands on the edge of a cliff and tosses a pebble into the ocean below. The graph of the height of the pebble versus time, h(t), is given below.
The function h is an example of what we refer to as a quadratic function, and is the next class of functions we will investigate. The types of questions we would like to answer regarding Tommy’s pebble include: how high was the cliff, how high did the pebble travel before it began to fall back down, when did it reach its highest point, and when did it hit the ground? We will return to solving these questions in future handouts when we have built enough skills and familiarity with quadratic functions.
Algebra II
Q1 Algebra Review 2 Handouts Page 1
Algebra Review 2 – Quadratic Functions Introduction 2.1 – Warmup 1 - Quadratic Function Standard Form
Name ______________________ Per ____ Date _______________
In previous lessons we were reminded that a linear function is a function that can be symbolically represented as f(x) = mx + b, where m and b are numbers and x is an independent variable. Note: m and b are not necessarily “nice” numbers. When we use the term “numbers” we are referring to the “real numbers”, which include the nice numbers such as integers (e.g. 2, 100, -3, etc.), but they also include rational numbers (e.g. 1 , 1 ,100 ), and irrational numbers 2
(e.g.
3
301
2 or π ).
Definition: A quadratic function is a function that can be symbolically represented as f(x) = ax2 + bx + c , where a, b, and c are numbers and x is an independent variable, with the additional requirement that the coefficient “a” of x2 cannot equal zero. This is referred to as the Standard Form of a Quadratic Function. Reflection: Why do you think we require that “a” cannot equal 0?
For each function below, indicate if it represents a quadratic function or not. If it is not, explain why not. Is this a quadratic function? YES
NO
f(x) = -2x2 + 6x + 9 f(x) = 0x2 - x + 12 f(x) =
−23 2 x - 11x + 0 17
f(x) = x2 + 15 f(x) = -x2 + 2 x - 4 f(x) = 6 f(x) = 9 + 6x - 25x2 f(x) = 2x - 3
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.2 – Graph g(x) = ax2 + c
Name ______________________ Per ____ Date _______________
The easiest functions to understand and manipulate are Linear Functions. The next function in terms of increased difficulty is the Quadratic Function, a function that can be written in standard form as f(x) = ax2 + bx + c. Notice, the standard form of a quadratic ends with a linear expression, bx + c and begins with a purely quadratic expression, ax2, as a summand. Just as linear functions are easily recognizable as expressions that can be algebraically manipulated into the form mx + b, so too will it be easy to recognize expressions that can be manipulated into this standard form, and hence be recognized as quadratic functions. During the next lessons we will learn how to symbolically manipulate quadratic functions to gain information, how to graph them from their various symbolic forms (yes, there are two additional symbolic forms other than the standard form), and which symbolic form is the easiest to use under varying circumstances. The simplest quadratic function, the one we refer to as the “Parent Function” for quadratic functions, is the function defined as f(x) = x2. 1. Fill in the following tables of values for f(x) = x2 and use your results to graph f(x) = x2. x 0 1 2 3 4
f(x) = x2
x -1 -2 -3 -4
f(x) = x2
x ½ 1/3 ¼
f(x) = x2
The graph of f(x) = x
2
is called a parabola (u-shaped graph).
f(x) = x2 is what we call an even function because the graph has the same positive heights on both the left and right side of the y-axis making the y-axis a line of symmetry. In other words, the height at any negative xvalue is the same as the height at the corresponding positive x-value. The lowest point is called the vertex.
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.2 – Graph g(x) = ax2 + c
Name ______________________ Per ____ Date _______________
Notes: We will show in a later lesson that the graph of every quadratic function is a parabola. You may assume this fact as you proceed. The Domain (set of inputs) for the parent function is the set of real numbers, since f(x) = x2 is defined for all numbers. The domain for every quadratic function is the set of real numbers. The Range (set of outputs) is restricted to all non-negative real numbers since squaring an input never results in a negative number. Another way of thinking of this property is that for each nonnegative number there is a point on the graph of f(x) = x2 with that height. In fact, for each positive number there are two points on the graph with that height. The range will vary for different quadratic functions. 2. Fill in the following tables of values for functions of the form f(x) = ax2 for various values of the leading coefficient “a”. x 0 1 2 3 4
k(x) = 2x2 x 0 1 2 3 4
g ( x) =
1 2 x 2
x 0 1 2 3 4
h(x) = -2x2
3. Graph k, g, and h above on the same coordinate axes below. Notice, f(x) = x2 is already included.
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.2 – Graph g(x) = ax2 + c
Name ______________________ Per ____ Date _______________
4. Reflections: Describe the effect that different values of “a” have on the graph in comparison to that of the parent function f(x) = x2. If a > 1 then ______________________________________________________ If 0 < a < 1 then __________________________________________________ If a < 0 then ______________________________________________________ 5. Fill in the following tables of values for functions of the form f(x) = ax2 + c for various values of the leading coefficient “a” and the constant term “c”. x 0 1 2 3 4
6.
G(x) = x2 + 1
x 0 1 2 3 4
H(x) = 2x2 - 1
x 0 1 2 3 4
K(x) = -2x2 + 3
Graph G, H, and K above on the same coordinate axes below. Notice, f(x) = x2 is already included.
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.2 – Graph g(x) = ax2 + c
Name ______________________ Per ____ Date _______________
7. Reflections: What is the role of the constant term “c” in the quadratic function f(x) = ax2 + c in terms of its effect on the graph of the quadratic function defined by g(x) = ax2? How did adding “c” change the range?
Making a table of values is one way to graph a quadratic function, but in most instances it is much easier to simply identify the effect that symbolic changes have on the parent function. Summary of the effects different values of a and b have on the graph of g(x) = ax2 + c: i. Changing the value of “a” in g(x) = ax2 simply increases all heights on the parent function defined by f(x) = x2 by a factor of “a”, provided “a” is positive. Thus, g(x) = 3x2 will appear to grow faster than the parent function and h(x) = ½ x2 will appear to grow slower than the parent function. The vertex remains the same. ii. Negative values of “a” will reflect the parent function across the x-axis, in which case it will open downward instead of upward. The vertex remains the same, but it is now a maximum instead of a minimum. iii. Adding a non-zero number “c” to g(x) = ax2 will shift the function up “c” units if “c” is positive and shift it down if “c” is negative. The x-value of the vertex remains zero, but the y-value of the vertex is shifted up or down “c” units.
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.2 – Graph g(x) = ax2 + c
Name ______________________ Per ____ Date _______________
8. Quickly sketch graphs of the following quadratic functions. You have 10 minutes for this exercise, so concentrate only on the most important aspects of the graph in comparison to the parent function. The graph of the parent function f(x) = x2 is provided.
g ( x ) = 2x2 −1
h ( x ) = − x2 + 2
What is the range of g?
What is the range of h?
m ( x ) = 1 x2 + 3 2
n ( x ) = 3x 2 − 2
What is the range of m?
k ( x ) = −2 x2 − 1
What is the range of k?
Algebra II
What is the range of n?
p ( x ) = − 1 x2 −1 4
What is the range of p?
Q1 Algebra Review 2 Handouts Page 7
Algebra Review 2 – Quadratic Functions Introduction 2.2 – Graph g(x) = ax2 + c
Name ______________________ Per ____ Date _______________
x and y intercepts 9. Recall: a y – intercept of a function is the point where the function crosses the y – axis. An x – intercept(s) of a function is a point where the function crosses the x – axis. a. Explain why a function can have at most one y – intercept.
b. Explain how to find the x and y values for the y – intercept of a function given in symbolic form.
c. Explain why a function can have more than one x-intercept and graph a function that has more than one x – intercept.
d. Explain how to find the x and y values for all x – intercepts for a function defined in symbolic form.
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.2 – Graph g(x) = ax2 + c
Name ______________________ Per ____ Date _______________
10. Graph g(x) = x2 – 4.
Graphically identify both coordinates of the x – intercept(s) by circling them.
Algebraically compute the x – intercepts. Do your answers match?
Repeat this process for the y – intercept.
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.2 – Graph g(x) = ax2 + c
Name ______________________ Per ____ Date _______________
11. Each of the following are graphs of functions of the form g(x) = ax2 + c. Identify the values of “c” for each, and whether a > 0 or a < 0.
c = ____________ “a” is ______________
c = ____________ “a” is ______________
c = ____________ “a” is ______________
c = ____________ “a” is ______________
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.3 – Homework 1: g(x) = ax2 + c
Name ______________________ Per ____ Date _______________
1. How does the graph of the quadratic function defined by f(x) = 3x2 compare to the graph of the quadratic function defined by g(x) = 2x2?
2. Identify the vertex for g(x) = ax2 + c for various values of “a” and “c”. a. If a = 2 and c = 3, then the vertex is located at the point _____________________ b. If a = -2 and c = 3, then the vertex is located at the point ____________________ c. If a = ½ and c = 3, then the vertex is located at the point ____________________ d. If a = - ½ and c = -3, then the vertex is located at the point _________________ 3. Recall that the range of the function defined by f(x) = x2 is all nonnegative numbers. a. What is the range of the function defined by g(x) = x2 + 2?
b. What is the range of the function defined by h(x) = x2 – 3?
c. What is the range of the function defined by k(x) = 2x2?
d. What is the range of the function defined by m(x) = 2x2 + 3?
e. What is the range of the function defined by n(x) = -x2?
f. What is the range of the function defined by n(x) = -x2 +1?
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.4 – Laws of Exponents Review
Name ______________________ Per ____ Date _______________
Laws of Exponents: (Assume x is a number in each case below.) Multiplication Rule: x n x m = x n + m (e.g. x 2 x3 = ( xx)( xxx) = xxxxx = x5 = x 2+3 ) Division Rule:
xn = x n−m m x
⎛ ⎞ x5 xxxxx e . g . = = x 2 = x5−3 ⎟ ⎜ 3 x xxx ⎝ ⎠ n
( )
Exponentiation Rule: x m
(
3
= x m*n e.g. ( x 2 ) = x 2 * x 2 * x 2 = ( xx)( xx)( xx) = xxxxxx = x6 = x 2*3
)
Zero Exponent Rule: x 0 = 1 , provided x is not zero. This follows because x0 * x = x0+1 = x . Negative Exponent Rule: x −1 =
1 This follows because x −1 * x1 = x −1+1 = x0 = 1. x
Important Algebra Rules: Assume A, B, C, a, b, and c are numbers. If AB = 0 then either A = 0, B = 0, or both A and B are 0. A(B+C) = AB + AC (Distributive Law) Therefore, since (a + b) resembles A in this definition, (a + b)(c + d) = (a + b)c + (a + b)d = ac + bc + ad + bd. This is sometimes referred to as FOIL, which stands for First, Outside, Inside, and Last as seen in the diagram below, but it is simply the distributive law applied twice. First
FOIL
Last
(a + b)(c + d) Inside Outside
Notice that this product represents the sum of the four possible products resulting from one term from the first factor and one from the second. Another way of viewing the FOIL process is to begin with “a” and create all possible multiples of “a” with a term from the second factor, namely “c” and “d” resulting in ac + ad. You then repeat this process with the other first factor term “b”.
In Algebra II and beyond, it is generally the case that at least one of the terms in each factor involves a variable, such as x, and that some of the terms resulting from FOIL can be combined as like terms. Below is an example. (x !1)(x + 2) = x 2 + 2x ! x ! 2 = x 2 + x ! 2 F O I L
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.4 – Laws of Exponents Review
Name ______________________ Per ____ Date _______________
Perform the following operations and be sure to combine like terms and do not leave negative exponents in your answers.
1. (3x 2 )(4 x 4 ) = _______________
2.
2 x4 = _______________ 4 x3
3.
10 x3 = _______________ 2 x5
4. (2 x3 )3 = _______________ 5. x( x − 1) = _______________ 6. ( x − 1)( x + 1) = _______________ 7. ( x − 1)(2 x + 1) = _______________ 8. ( x − 1)( x − 5) = _______________ 9. ( x − 1)2 = _______________ (Careful! A2 = A*A)
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.5 – Homework 2 – Laws of Exponents Review
Name ______________________ Per ____ Date _______________
Expand the following expressions (FOIL) and whenever possible write your answers as quadratic expressions in standard form (ax2 + bx + c).
1. ( x + 2)(2 − 3x) = _______________ 2. −2( x + 4)( x − 5) = _______________ 3. (2 − x)(8 − 2 x) = _______________ 4. ( x − 8)( x + 3) = _______________ 5. (2 x − 3)(5 x − 6) = _______________ 6. ( x + 2)( x − 2) = _______________ 7. ( x + 3)( x − 3) = _______________
1 ⎞⎛ 1 ⎞ ⎛ 8. ⎜ x + ⎟⎜ x − ⎟ = _______________ 2 ⎠⎝ 2 ⎠ ⎝ 9. x( x − 1)( x + 1) = _______________
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.6 – Warmup 2 – Solving Equations
Name ______________________ Per ____ Date _______________
One of the more important challenges throughout Algebra is to solve equations. As you are exposed to an ever increasing variety of functions it is important to identify the type of equation you are attempting to solve because the techniques used to solve equations is different depending on the type of equation. Note: equations are the equivalent to “questions” in English grammar. Do not confuse them with functions, which also include an equal sign; but functions are not questions. Functions are statements that define collections of points (input/output pairs). Associated with this collection of points are questions. For example, associated with the linear function defined by f(x) = 2x – 1 we might ask “for which x is f(x) equal to 3”? Problem 1 below answers that question. Solve the following linear equations: 1. 2x – 1 = 3
x = ______________________________
2. − x − 1 = −2
x = ______________________________
You will learn a variety of techniques for solving certain types of quadratic equations. In previous lessons, while trying to locate x – intercepts you were required to solve quadratic equations of the form x2 – c = 0. You used balancing equation techniques to first rewrite this equation as x2 = c, and then used the square root to solve. Example 1:
Solve x2 – 8 = 1. x2 = 9 x = ±3
add positive 8 to both sides since 32 = 9 and (-3)2 = 9
Quadratic equations in other certain forms are also very easy to solve. Definition: A quadratic function in factored form is one that is written as f ( x) = a( x − s)( x − t ) , where a, s, and t are [possibly negative] real numbers, a not equal to zero. Note: all of the problems in 2.5 – Homework 2 were quadratic expressions written in factored form. Of particular interest is finding for which x such expressions have output 0. Note: many of your answers to the previous homework problems included a non-zero x-term. Solving quadratic equations that have a non-zero x-term are particularly difficult. Of those quadratic equations that include a non-zero linear x-term, the easiest to solve are those in factored form. An equation in factored form is one that can be written as A * B = 0, where A and B are [possibly] variable expressions. (e.g. expressions 1 – 8 in the previous homework.)
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.6 – Warmup 2 – Solving Equations
Name ______________________ Per ____ Date _______________
Example 2: Solve (x – 1)(x + 3) = 0. Notice, this equation is of the form A*B = 0, where A = (x – 1) and B = (x + 3). We should recall that according to the Zero-Product Property, if A*B = 0, then either A = 0 or B = 0. Thus, either x – 1 = 0 or x + 3 = 0. So, by using the Zero-Product Property we can reduce our quadratic equation to two linear equations. And, we know that linear equations can be easily solved just by applying balancing-equation techniques. There are two solutions to this quadratic equation, each one corresponding to one of the factors. x – 1 = 0 implies that x = 1. x + 3 = 0 implies that x = -3 (be careful, our answers are not 1 and 3, but 1 and negative 3). Use the Zero-Product Property to solve the following quadratic equations presented in factored form. Show all work! 3. (x – 4)(x + 10) = 0
x = ______________________________________
4. (x + ¼ )(x + 9) = 0
x = ______________________________________
5. 2(x – 4)(x + 10) = 0
x = ______________________________________
6. 2x(x + 10) = 0
x = ______________________________________
7. (x – 40)(x + 100) = 0
x = ______________________________________
Reflection: Why would it be difficult to use a table of values to solve the above?
Reflection: Why can’t you use the Zero-Product Property to solve (x – 1)(x + 3) = 1?
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.7 – Graphing from Factored Form
Name ______________________ Per ____ Date _______________
Below are the graphs for two quadratic functions, each with the x and y intercepts clearly marked. Note: the units for both graphs are NOT necessarily the same.
1. Since the units are not marked, it is impossible to identify both coordinates of each point, but it is always possible to identify one of the coordinates for an intercept. Fill in one coordinate for each of the points below. a. P = (
,
)
d. p = (
,
)
b. Q = (
,
)
e. q = (
,
)
c. R = (
,
)
f. r = (
,
)
2. Reflection: Fill in the following blanks: the _________ coordinate of a y – intercept is always ________. The _________ coordinate of an x – intercept is always ________.
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.7 – Graphing from Factored Form
Name ______________________ Per ____ Date _______________
3. Reflection: If you were given an algebraic formula for the quadratic function f above, how would you find the unknown coordinate for P? How would you find it for R?
4. The formula for the function f on the previous page is f(x) = 2(x – 1)(x + 3). Use this formula and your results from problem 3 to find the missing coordinates of the x – intercepts. Use it to find the missing coordinate for the y – intercept. Label the tick marks on each axis for the graph of f with an appropriate scale (i.e. each tick mark represents what quantity for each axis?).
Reflections: What was the connection between the two factors, (x – 1) and (x + 3) and the x – intercepts?
What equation did you need to solve in order to locate the x – intercepts?
How was the process different for finding the missing coordinate for the y – intercept?
Definition The x-coordinate of an x-intercept is referred to as a zero of f. Careful, there is an obvious connection between x – intercepts and zeros, but the x – intercept is a point and the zero is a number, so they are not the same. 5. Fill in the following blanks: In order to find the zeros for a quadratic function defined in the form f(x) = a(x + s)(x + t), first set ________ = 0 and solve for x. This requires setting each factor to zero and solving each for x. The y-coordinate of the x – intercepts equals _______. In order to find the y-coordinate of the y-intercept all you need to do is to evaluate _________ at ________. The x-coordinate of the y – intercept is always _________.
6. The formula for the function g on the previous page is g(x) = – (x – 1)(x -5). Use this formula to find the missing coordinate of each x – intercept. Use it to find the missing coordinate of the y – intercept. Finally, label the tick marks on each axis for g with an appropriate scale. Show all work!
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.7 – Graphing from Factored Form
Name ______________________ Per ____ Date _______________
7. What is the relationship between the zeros for the previous function f and the location of its vertex? Use your answers to problem 4 to find the x – coordinate of the vertex. Use your x–coordinate to then find the y–coordinate. Is your algebraic answer consistent with what you view on the graph?
8.
Use the graph of g to approximate its vertex. Follow the procedure you used in the previous problem to algebraically find the exact location of the vertex.
9. Graph the function defined by f ( x) = ( x − 1)2 . Hint: f can be rewritten as f ( x) = ( x − 1)(x − 1) .
Reflection: How do graphs of functions in the form f ( x) = a( x − h)2 differ from ones of the form f ( x) = a( x − s)( x − t ) where s ≠ t ?
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.7 – Graphing from Factored Form
Name ______________________ Per ____ Date _______________
10. Suppose f(x) = 2(x – 2)(x – 4) and the graph of the quadratic function g is given below. Which has the smaller minimum value, f or g? Explain
Summary – What we should know thus far about the graph of f ( x) = a( x − s)( x − t ) : − Since f is a quadratic, its graph is a parabola. − In order to find its x – intercepts, set each factor equal to 0, resulting in x – intercepts at the points (s, 0) and (t, 0). (Caution: either s or t may be negative, in which case the factors may look like (x + s) or (x + t), which yield negative x-coordinates, or s and t may be equal to one another, in which case the vertex lies on the x-axis and there is only one x-intercept.) − In order to find the y – intercept, simply evaluate the function at x = 0. The y – intercept will be located at (0, f(0)), as with any function. − If a > 0 then the parabola will open upwards. − If a < 0 then the parabola will open downwards. − The x – coordinate of the vertex is located halfway between the x – intercepts. − The y – coordinate of the vertex can be found by evaluating f at the x – coordinate.
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.7 – Graphing from Factored Form
Name ______________________ Per ____ Date _______________
Note: another way of thinking about the relationship between factors of a quadratic function and its zeros is the following: factors yield zeros (simply set each factor to zero and solve for x). Example: Graph the quadratic function defined by f(x) = 2(x-1)(x+5). Be sure to indicate both coordinates of each intercept and the vertex. Solution: f is a quadratic, so its graph is a parabola. a.
Find the zeros: Zeros occur at points where f(x) = 0 (x-intercepts on the graph) Set 2(x – 1)(x+5) = 0. By the Zero-Product Property this implies either x - 1 = 0 or x + 5 = 0, which implies x = 1 or x = -5.
b. Using these zeros, locate the x-intercepts on the graph at the points (1, 0) and (-5, 0). c. Using the fact that a = 2, we know that the parabola opens upward. d. Find the x-coordinate of the vertex, which is located halfway between the two zeros. The x-coordinate of the vertex is therefore -2 (the midpoint between 1 and -5 can be found by guess and check or you can compute it by adding 1 and -5 and dividing by 2). e. The vertex is therefore located at the point (-2, f(-2)). f(-2) = 2(-2 – 1)(-2 + 5) = 2(-3)(3) = -18. Locate the vertex at (-2, -18). f. Find the y-intercept: The y-intercept is located at the point (0, f(0)). f(0) = 2(0 - 1)(0 + 5)=2(-1)(5) = -10. Locate the y-intercept at the point (0, -10). g. Graph f. Notice, this is the same as graphing f(x) = 2(x2 + 4x – 5) = 2x2 + 8x – 10, but you can see that it is much easier graphing f in its factored form.
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.8 – Exit Pass
Name ______________________ Per ____ Date _______________
1. Graph the function defined by f(x) = (x - 3)(x – 5).
Algebra II
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Algebra Review 2 – Quadratic Functions Introduction 2.9 – Homework 3
Name ______________________ Per ____ Date _______________
Graph the following quadratic functions. Be sure to include both coordinates of all intercepts and the vertex. 1. Graph the function defined by f(x) = (x +2)(x – 2)
2. Graph the function defined by f(x) = - (x +2)(x – 1)
3. Graph the function defined by f(x) = -½ (x +1)(x – 3)
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