Lesson 3.1 Skills Practice Name_________________________________________________________ Date__________________________
Hitting the Slopes Determining Rate of Change from a Graph
Vocabulary Match each definition to its corresponding term. a. rate of change
1. rate which has a second term of 1 unit e. unit rate 2. ratio in which the units of the quantities being compared are different
b. per
d. rate 3. the vertical change from one point to another point on a graph
c. run
g. rise 4. phrase used when a rate is used to describe a rate of increase
d. rate
(or decrease) in a real-life situation a. rate of change 5. the rate of change on a graph written as the vertical change from one
e. unit rate
point to another over the horizontal change of the same two points
____
© 2011 Carnegie Learning
f. rise run rise f. ____ run
6. means “for each” or “for every” b. per 7. the horizontal change from one point to another point on a graph
g. rise
c. run
Chapter 3 Skills Practice • 409
Lesson 3.1 Skills Practice
page 2
Problem Set Write a rate for each situation. State whether the rate is a rate of increase or a rate of decrease. 1. Use the graph to write a rate that compares the track length to the change in time at point A. y
Length of Track Remaining in Bobsled Run
600 540
Track length (in meters)
A 450 360 B 270 180
C
90
0
2
4
6
8
10
12
14
16
18
20
x
Time (in seconds)
Point A is at (4, 480). Because the bobsled started at 600 meters, the change in the track length remaining is 120 meters. The change in time is 4 seconds. So, the rate of decrease is 120 . 4
© 2011 Carnegie Learning
____
410 • Chapter 3 Skills Practice
Lesson 3.1 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ 2. Use the graph to write a rate that compares the track length to the change in time at point B. y
Length of Track Remaining in Bobsled Run
600 540
Track length (in meters)
A 450 360 B 270 180
C
90
0
2
4
6
8
10
12
14
16
18
20
x
Time (in seconds)
Point B is at (10, 300). Because the bobsled started at 600 meters, the change in the track length remaining is . 300 meters. The change in time is 10 seconds. So, the rate of decrease is 300 10
© 2011 Carnegie Learning
____
Chapter 3 Skills Practice • 411
Lesson 3.1 Skills Practice
page 4
3. Use the graph to write a rate that compares the track length to the change in time at point C. y
Length of Track Remaining in Bobsled Run
600 540
Track length (in meters)
A 450 360 B 270 180
C
90
0
2
4
6
8
10
12
14
16
18
20
x
Time (in seconds)
Point C is at (15, 150). Because the bobsled started at 600 meters, the change in the track length remaining is . 450 meters. The change in time is 15 seconds. So, the rate of decrease is 450 15
© 2011 Carnegie Learning
____
412 • Chapter 3 Skills Practice
Lesson 3.1 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ 4. Use the graph to write a rate that compares the change in elevation to the change in time at point D. y
Distance Ascended by a Mountain Climber
200 F
180
Distance (in feet)
160 140 E
120 100 80 60
D
40 20 0
2
4
6
8
10
12
14
16
18
20
x
Time (in minutes)
Point D is at (5, 50). Because the mountain climber started at 0 feet, the change in elevation is 50 feet. The change in time is 5 minutes. So, the rate of increase is 50 . 5
© 2011 Carnegie Learning
___
Chapter 3 Skills Practice • 413
Lesson 3.1 Skills Practice
page 6
5. Use the graph to write a rate that compares the change in elevation to the change in time at point E. y
Distance Ascended by a Mountain Climber
200 F
180
Distance (in feet)
160 140 E
120 100 80 60
D
40 20 0
2
4
6
8
10
12
14
16
18
20
x
Time (in minutes)
Point E is at (12, 120). Because the mountain climber started at 0 feet, the change in elevation is 120 feet. The change . in time is 12 minutes. So, the rate of increase is 120 12
© 2011 Carnegie Learning
____
414 • Chapter 3 Skills Practice
Lesson 3.1 Skills Practice
page 7
Name_________________________________________________________ Date__________________________ 6. Use the graph to write a rate that compares the change in elevation to the change in time at point F. y
Distance Ascended by a Mountain Climber
200 F
180
Distance (in feet)
160 140 E
120 100 80 60
D
40 20 0
2
4
6
8
10
12
14
16
18
20
x
Time (in minutes)
Point F is at (18, 180). Because the mountain climber started at 0 feet, the change in elevation is 180 feet. The change . in time is 18 minutes. So, the rate of increase is 180 18
© 2011 Carnegie Learning
____
Chapter 3 Skills Practice • 415
Lesson 3.1 Skills Practice
page 8
Label two points on each graph to determine the rate of change for the graph. Write the rate in the format rise run and as a unit rate.
____
7.
8.
y 50
180
45
160
40
140
35
Elevation (in meters)
Elevation (in meters)
y 200
120 100
A
80 60
B
40
30 B
25 20 15 10
20 0
A
5 2
4
6
8
10 12 14 16 18 20
Time (in seconds)
x 0
1
2
3
4
Sample answer:
Point A: (4, 90)
Point A: (2, 35)
Point B: (6, 50)
Point B: (4, 25)
____ ___________ unit rate 5 ___________ 220 meters 1 second
6
7
8
9
10
x
Time (in seconds)
Sample answer:
____ rise 5 ___________ 210 meters
run
2 seconds
unit rate 5 __________ 25 meters 1 second © 2011 Carnegie Learning
240 meters rise run 5 2 seconds
5
416 • Chapter 3 Skills Practice
Lesson 3.1 Skills Practice
page 9
Name_________________________________________________________ Date__________________________ 9.
10.
y 60
y 200
54 150
42
Distance (in miles)
Distance (in feet)
48
B
36 30 A
24
B 100
A
18 50
12 6 0
2
4
6
8
10 12 14 16 18 20
Time (in minutes)
x
0
1
2
3
4
Sample answer:
Point A: (4, 24)
Point A: (6, 90)
Point B: (6, 36)
Point B: (8, 120)
____ __________ 6 feet unit rate 5 _________ 1 minute
6
7
8
9
10
x
Time (in hours)
Sample answer:
____ ________ unit rate 5 ________ 15 miles
30 miles rise run 5 2 hours 1 hour
© 2011 Carnegie Learning
12 feet rise run 5 2 minutes
5
Chapter 3 Skills Practice • 417
Lesson 3.1 Skills Practice
11.
12.
y
y
60
300
54
270
48
240
42
Distance (in miles)
Elevation (in meters)
page 10
A
36 30 24 18
210 180 150
A
120 90
B
12
60
6 0
B
30 2
4
6
8
10 12 14 16 18 20
Time (in seconds)
x
0
1
2
Sample answer:
Point A: (2, 36)
Point A: (2, 150)
Point B: (4, 12)
Point B: (4, 300)
____ ___________ unit rate 5 ___________ 212 meters 1 second
4
5
x
Time (in hours)
Sample answer:
____ _________ unit rate 5 ________ 75 miles
150 miles rise run 5 2 hours 1 hour
© 2011 Carnegie Learning
224 meters rise run 5 2 seconds
3
418 • Chapter 3 Skills Practice
Lesson 3.1 Skills Practice
page 11
Name_________________________________________________________ Date__________________________ Determine the unit rate of change for lines A and B on each graph. 13.
y 10
A
B
9 8
Cost (in dollars)
7 6 5 4 3
F
2 1
D
E 0 C 1
2
3
4
5
6
7
8
9
10
x
Number of items
Line A sample answer:
Line B sample answer:
Point C: (0, 0)
Point E: (0, 0)
Point D: (1, 1.5)
Point F: (1, 3)
____ _________ unit rate 5 _________ 3 dollars
3 dollars rise run 5 1 item 1 item
© 2011 Carnegie Learning
____ rise 5 __________ 1.5 dollars run 1 item unit rate 5 __________ 1.5 dollars 1 item
Chapter 3 Skills Practice • 419
Lesson 3.1 Skills Practice
14.
page 12
y 20
A
B
18 16
Cost (in dollars)
14
F
12 10
E D
8 C
6 4 2 0
1
2
3
4
5
6
7
8
9
10
x
Number of items
Line A sample answer:
Line B sample answer:
Point C: (1, 7)
Point E: (1, 9)
Point D: (2, 9)
Point F: (2, 13)
____ _________ 2 dollars unit rate 5 _________
2 dollars rise run 5 1 item
1 item
© 2011 Carnegie Learning
1 item
____ _________ unit rate 5 _________ 4 dollars
4 dollars rise run 5 1 item
420 • Chapter 3 Skills Practice
Lesson 3.1 Skills Practice
page 13
Name_________________________________________________________ Date__________________________ 15.
y 40
C
Elevation (in meters)
30
E
D
20
10 F B 0
2
4
6
A 8
10 12
14
16
18
20
x
Time (in seconds)
Line A sample answer:
Line B sample answer:
Point C: (2, 30)
Point E: (2, 24)
Point D: (4, 20)
Point F: (4, 8)
____ ___________ unit rate 5 __________ 25 meters 1 second
____ ___________ 28 meters unit rate 5 __________ 216 meters rise run 5 2 seconds 1 second
© 2011 Carnegie Learning
210 meters rise run 5 2 seconds
Chapter 3 Skills Practice • 421
Lesson 3.1 Skills Practice
16.
page 14
y 100 90
Elevation (in meters)
80 C
70 60
E
D
50 40 30
F
20 10
B 0
1
2
3
4
A 5
6
7
8
9
10
x
Time (in seconds)
Line A sample answer:
Line B sample answer:
Point C: (1, 70)
Point E: (1, 60)
Point D: (3, 50)
Point F: (3, 20)
____ ___________ unit rate 5 ___________ 210 meters 1 second
____ ___________ unit rate 5 ___________ 220 meters 240 meters rise run 5 2 seconds 1 second
© 2011 Carnegie Learning
220 meters rise run 5 2 seconds
422 • Chapter 3 Skills Practice
Lesson 3.1 Skills Practice
page 15
Name_________________________________________________________ Date__________________________ 17.
y 40
A
Cost (in dollars)
30
20
D B
10 F C 0 E
1
2
3
4
5
x
Number of items
Line A sample answer:
Line B sample answer:
Point C: (0, 0)
Point E: (0, 0)
Point D: (2, 18)
Point F: (2, 6)
____ __________ unit rate 5 _________ 9 dollars 1 item
____ _________ unit rate 5 _________ 3 dollars
6 dollars rise run 5 2 items 1 item
© 2011 Carnegie Learning
18 dollars rise run 5 2 items
Chapter 3 Skills Practice • 423
Lesson 3.1 Skills Practice
18.
page 16
y 20 18 C
16
Elevation (in feet)
14
D
E
12 10 F
8
A
6 4 2 B 0
2
4
6
8
10
12
14
16
18
20
x
Time (in minutes)
Line A sample answer:
Line B sample answer:
Point C: (4, 16)
Point E: (2, 13)
Point D: (8, 14)
Point F: (4, 8)
____ __________ 20.5 feet unit rate 5 _________ 1 minute
____ __________ unit rate 5 _________ 22.5 feet 1 minute 25 feet rise run 5 2 minutes
© 2011 Carnegie Learning
22 feet rise run 5 4 minutes
424 • Chapter 3 Skills Practice
Lesson 3.2 Skills Practice Name_________________________________________________________ Date__________________________
At the Arcade Determining Rate of Change from a Table
Vocabulary Define the term in your own words. 1. first differences First differences are the values determined by subtracting consecutive y-values in a table when the x-values are consecutive integers.
Problem Set Use the informal method to determine the rate of change for the data in each table. The rate of change is constant for the data in each table. Write the rate as a unit rate.
© 2011 Carnegie Learning
1.
Sample answer:
Number of Balloons
Total Cost of Balloons (in Dollars)
2
6
4
12
Cost of balloons 5 12 2 6
6
18
8
24
Number of balloons 5 4 2 2
52
56
__________ Unit rate 5 _________ 3 dollars 1 balloon
Rate of change 5 6 dollars 2 balloons
Chapter 3 Skills Practice • 425
Lesson 3.2 Skills Practice
2.
Number of Lawns
Total Earned (in Dollars)
3
25.50
5
42.50
7
59.50
9
76.50
page 2
Sample answer: Number of lawns 5 5 2 3
52 Total earned 5 42.50 2 25.50 5 17
__________ 8.5 dollars Unit rate 5 __________
Rate of change 5 17 dollars 2 lawns
Number of Touchdowns
Total Points Scored
2
12
3
18
4
24
5
30
Sample answer: Number of touchdowns 5 3 2 2 5 1 Total points scored 5 18 2 12
56
_____________ 6 points Unit rate 5 _____________ 1 touchdown
6 points Rate of change 5 1 touchdown
426 • Chapter 3 Skills Practice
© 2011 Carnegie Learning
3.
1 lawn
Lesson 3.2 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ 4.
Total Number of Calories Burned
15
180
30
360
Number of calories 5 360 2 180
45
540
60
720
Number of Hours
Total Number of Miles Traveled
5.
© 2011 Carnegie Learning
Sample answer:
Number of Minutes on an Exercise Bike
2
130
5
325
8
520
11
715
Number of minutes 5 30 2 15 5 15
5 180
____________ Unit rate 5 ___________ 12 calories 1 minute
Rate of change 5 180 calories 15 minutes
Sample answer: Number of hours 5 5 2 2 53 Number of miles 5 325 2 130 5 195
_________ Unit rate 5 ________ 65 miles
Rate of change 5 195 miles 3 hours 1 hour
Chapter 3 Skills Practice • 427
Lesson 3.2 Skills Practice
6.
page 4
Sample answer:
Number of Songs Downloaded
Total Cost of Songs (in Dollars)
10
9.90
20
19.80
Cost of songs 5 19.80 2 9.90
30
29.70
40
39.60
Number of songs 5 20 2 10 5 10
5 9.9
_________ $0.99 Unit rate 5 _______ 1 song
$9.90 Rate of change 5 10 songs
_______
y 2y Use the formula x 2 2 x 1 to calculate the unit rate of change for the data in each table. The rate of 2
1
change is constant for the data in each table.
Number of Raffle Tickets
Total Cost of Raffle Tickets (in Dollars)
2
1
4
2
8
4
10
5
Sample answer: ( x1, y1 ) 5 (2, 1) ( x2, y2 ) 5 (4, 2) y 2y ________ 5 ______ 2 2 1 422 x 2x 5 __ 1 2 0.5 5 ___ 1 $0.50 Unit rate 5 _______ 2
2
1
1
1 ticket
428 • Chapter 3 Skills Practice
© 2011 Carnegie Learning
7.
Lesson 3.2 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ 8.
Sample answer: x
y
22
8
0
0
2
28
4
216
( x1, y1 ) 5 (22, 8) ( x2, y2 ) 5 (2, 28)
________ ________
y 2 y1 2 5 28 2 8 x2 2 x1 2 2 (22)
_____ 5 ___ 24
5 216 4 1
___
Unit rate 5 24 1
© 2011 Carnegie Learning
9.
Sample answer:
Number of Photos Printed
Total Cost of Photos (in Dollars)
10
2
( x2, y2 ) 5 (20, 4)
20
4
y 2 y1 2 5 4 2 2 x2 2 x1 20 2 10
30
6
40
8
5 2 10
( x1, y1 ) 5 (10, 2)
________ ________
___ 5 ___ 0.2 1 $0.20 Unit rate 5 ________ 1 photo
Chapter 3 Skills Practice • 429
Lesson 3.2 Skills Practice
10.
page 6
Sample answer:
x
y
3
27
( x1, y1 ) 5 (3, 27)
5
45
( x2, y2 ) 5 (5, 45)
7
63
9
81
________ ________
y 2 y1 45 2 27 2 5 x2 2 x1 523
___ 5 __ 9
5 18 2 1
__
Unit rate 5 9 1
Sample answer:
Number of Greeting Cards
Total Cost of Greeting Cards (in Dollars)
2
6.50
( x2, y2 ) 5 (3, 9.75)
3
9.75
y 2 y1 9.75 2 6.50 2 5 x2 2 x1 322
6
19.50
8
26.00
( x1, y1 ) 5 (2, 6.50)
________ ___________
_____ $3.25 Unit rate 5 ______
5 3.25 1
1 card
430 • Chapter 3 Skills Practice
© 2011 Carnegie Learning
11.
Lesson 3.2 Skills Practice
page 7
Name_________________________________________________________ Date__________________________
12.
x
y
Sample answer:
23
54
( x1, y1 ) 5 (23, 54)
1
218
( x2, y2 ) 5 (1, 218)
3
254
7
2126
________ __________
y 2 y1 218 2 54 2 5 x2 2 x1 1 2 (23)
_____ 5 _____ 218 1 Unit rate 5 _____ 218 5 272 4
1
Calculate the rate of change between the points listed in each table. Determine if the table represents a linear relationship.
© 2011 Carnegie Learning
13.
(2, 14) and (5, 35)
(5, 35) and (7, 49)
x
y
2
14
5
35
49 2 35 ________ ________ 35 2 14 21 5 ___ 5 ___ 14 725 522 3 2 7 5 __ 7 5 __
7
49
(7, 49) and (10, 70)
10
70
Yes, the table represents a linear relationship.
1
1
________ 5 ___ 21 70 2 49 10 2 7 3 7 5 __ 1
Chapter 3 Skills Practice • 431
Lesson 3.2 Skills Practice
x
y
210
50
22
10
(210, 50) and (22, 10)
220 2 10 ___________ 240 5 _____ 230 5 _____ __________ 10 2 50 8 6 22 2 (210) 4 2 (22) 25 5 ___ 5 ___ 25 1
4
220
(4, 220) and (14, 270)
14
270
270 2 (220) 250 5 14 2 4 10
Yes, the table represents a linear
1
____________ _____ 5 ___ 25 1
relationship.
15.
(22, 10) and (4, 220)
(21, 224) and (2, 48) (2, 48) and (4, 90)
x
y
21
224
2
48
48 2 (224) ___ 90 2 48 ___________ ________ 42 5 72 5 ___ 3 422 2 2 2 (21) 21 5 ___ 24 5 ___
4
90
(4, 90) and (8, 192)
8
192
5 102 192 2 90 4 824
No, the table does not represent a linear relationship.
432 • Chapter 3 Skills Practice
1
1
_________ ____ 25.5 5 _____ 1 © 2011 Carnegie Learning
14.
page 8
Lesson 3.2 Skills Practice
page 9
Name_________________________________________________________ Date__________________________
16.
(26, 12) and (23, 6) (23, 6) and (3, 26)
x
y
26
12
23
6
3
26
(3, 26) and (6, 210)
6
210
210 2 (26) 24 5 623 3
26 2 6 __________ 5 ___ 26 5 _____ 212 ________ 6 2 12 3 6 23 2 (26) 3 2 (23) 22 5 ___ 5 ___ 22 1
1
___________ ___
No, the table does not represent a linear relationship.
© 2011 Carnegie Learning
17.
(2, 13.5) and (5, 33.75)
x
y
2
13.5
5
33.75
10
67.5
(5, 33.75) and (10, 67.5)
15
101.25
5 33.75 67.5 2 33.75 5 10 2 5
Yes, the table represents a linear relationship.
____________ 5 ______ 20.25 33.75 2 13.5 522
3
_____
5 6.75 1
____________ ______ 6.75 5 _____ 1
(10, 67.5) and (15, 101.25) 67.5 _____________ 5 ______ 33.75 101.25 2 5 15 2 10 5 _____ 6.75 1
Chapter 3 Skills Practice • 433
Lesson 3.2 Skills Practice
18.
page 10
(24, 238) and (21, 29.5)
x
y
24
238
21
29.5
29.5 2 (238) _____ _____________ 5 28.5 3 21 2 (24) 9.5 5 ___
2
19
(21, 29.5) and (2, 19)
3
27
No, the table does not represent a linear relationship.
1
19 2 (29.5) _____ ___________ 5 28.5 3 2 2 (21) 5 ___ 9.5 1
(2, 19) and (3, 27)
________ 27 2 19 5 __ 8 1
© 2011 Carnegie Learning
322
434 • Chapter 3 Skills Practice
Lesson 3.3 Skills Practice Name_________________________________________________________ Date__________________________
To Put It in Context Determining Rate of Change from a Context
Problem Set Determine the rate of change for each situation. 1. Lashawna is making jewelry to sell at a craft fair. On Monday, she makes 12 bracelets. On Tuesday, she works an additional 2.5 hours and has a total of 22 bracelets. What is the unit rate of the time it takes her to make each bracelet? 22 bracelets 2 12 bracelets 5 10 bracelets made on Tuesday
____________ 5 __________ 0.25 hour 2.5 hours 10 bracelets 1 bracelet The unit rate is __________ 0.25 hour or 15 minutes per bracelet. 1 bracelet 2. Nina and her friends are going to the downtown rib festival. The festival organizers expect 10,000 people to attend the four-day festival. At the end of the festival the organizers say that they have exceeded their expected attendance by 2000 people. What was the average number of people to attend the festival per day? 10,000 people 1 2000 people 5 12,000 people
______________ ____________
© 2011 Carnegie Learning
12,000 people 3000 people 5 4 days 1 day
____________
3000 people The unit rate is . 1 day
Chapter 3 Skills Practice • 435
Lesson 3.3 Skills Practice
page 2
3. Rosa is ordering a submarine sandwich from the corner deli. The deli charges $6.25 for a 7-inch sub. Some additional toppings cost extra. Rosa’s sandwich with two extra toppings costs $7.75. What is the cost per additional topping? $7.75 2 $6.25 5 $1.50 for extra toppings $0.75 $1.50 __________ 5 _________ 2 toppings 1 topping $0.75 The unit rate is _________ . 1 topping
4. Aiko spends 2.5 hours baking croissants for a community center bake sale. Aiko bakes the 90 croissants in 5 batches. What is the unit rate of the number of batches baked per hour?
__________ __________ The unit rate is __________ 2 batches .
5 2 batches 5 batches 2.5 hours 1 hour 1 hour
5. Nelson is selling his photographs at an art festival. The festival is open for 6 hours each day for the number of photographs sold per hour? 3 days 3 6 hours per day 5 18 hours total
_______________ ______________ 3 photographs The unit rate is ______________ . 1 hour 54 photographs 3 photographs 5 18 hours 1 hour
436 • Chapter 3 Skills Practice
© 2011 Carnegie Learning
3 days. At the conclusion of the festival, Nelson has sold 54 photographs. What is the unit rate of
Lesson 3.3 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ 6. Clayton wants to purchase tickets for the rides at a carnival. He can choose to purchase tickets individually or he can purchase a ticket package. The package includes 25 tickets for $18.75. What is the cost per ticket if he purchases the package?
__________ $0.75 5 ________ 1 ticket $0.75 The unit rate is _______ . $18.75 Package 5 25 tickets
1 ticket
7. Carmen is selling pies at the cherry festival to raise money for her local volunteer fire department. She sells 85 pies for $12 each. The supplies to make the pies cost Carmen $340. What is the unit rate of the profit made for each pie? 85 pies 3 $12 per pie 5 $1020 earned $1020 2 $340 5 $680 profit
_______ ______ $8 The unit rate is ______ .
$8 $680 5 85 pies 1 pie
© 2011 Carnegie Learning
1 pie
8. Tameca is planning a hiking trip. The trail she would like to follow is 7.5 miles long. She plans to start her hike at 10:00 am. She hopes to reach the end of the trail at 3:00 pm. What is the unit rate of the number of miles per hour that Tameca plans to hike? 10:00 am to 3:00 pm is 5 hours.
_________ _________ . The unit rate is _________ 1.5 miles
5 1.5 miles 7.5 miles 5 hours 1 hour 1 hour
Chapter 3 Skills Practice • 437
Lesson 3.3 Skills Practice
page 4
9. Jamal is shopping with a gift card he received for his birthday. After he purchases two T-shirts the gift card balance has dropped from $50 to $20.02. What is the unit rate of the cost per T-shirt? $50 2 $20.02 5 $29.98 spent $14.99 $29.98 _________ 5 ________ 2 T-shirts
1 T-shirt
________
$14.99 The unit rate is . 1 T-shirt
10. Olivia is printing photos for a scrapbook project. She prints 150 photos in 1 hour and 15 minutes. What is the unit rate of the number of minutes it takes to print each photo? 1 hour 5 60 minutes 60 minutes 1 15 minutes 5 75 minutes
___________ __________ The unit rate is __________ 0.5 minute . 1 photo
© 2011 Carnegie Learning
5 0.5 minute 75 minutes 150 photos 1 photo
438 • Chapter 3 Skills Practice
Lesson 3.3 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ 11. Franco is traveling to a vacation destination 715 miles from home. On the first day of his trip he travels 390 miles in 6 hours. On the second day of his trip he leaves at 8:00 am and arrives at his destination at 1:00 pm. What is the unit rate of the total number of miles per hour traveled? Convert y 2y to answer the question. the information to coordinate points and use the formula _______ x2 2 x1 2 1 8:00 am to 1:00 pm is 5 hours 6 hours 1 5 hours 5 11 hours (x1, y1) 5 (6, 390) (x2, y2) 5 (11, 715)
__________ 325 miles 5 _________ 5 hours 5 ________ 65 miles 1 hour The unit rate is ________ 65 miles . y2 2 y1 715 2 390 _______ 5 x2 2 x1 11 2 6
© 2011 Carnegie Learning
1 hour
Chapter 3 Skills Practice • 439
Lesson 3.3 Skills Practice
page 6
12. Rakesha loves reading and is participating in a read-a-thon to raise money for a charity. She plans to read 15 books during the 90-day read-a-thon. During the first 30 days she reads 7 books. What is the unit rate of the number of days she has to read each book to meet her goal? Convert the y 2y to answer the question. information to coordinate points and use the formula _______ x2 2 x1 2 1 (x1, y1) 5 (7, 30) (x2, y2) 5 (15, 90)
_______ ________ 60 days 5 ________ y 2y x2 2 x1 5 90 2 30 15 2 7 2 1 8 books
________ 7.5 days . The unit rate is ________
7.5 days 5 1 book
© 2011 Carnegie Learning
1 book
440 • Chapter 3 Skills Practice
Lesson 3.4 Skills Practice Name_________________________________________________________ Date__________________________
All Together Now! Determining Rate of Change from an Equation
Vocabulary Give an example of each term. 1. slope Sample answer: The slope of the line represented by the equation y 5 3x 1 4 is 3. 2. slope-intercept form Sample answer: The equation y 5 3x 1 4 is written in slope-intercept form.
Problem Set
© 2011 Carnegie Learning
Determine the slope of the line represented by each equation. 1. y 5 4x 1 5 1 2x
2. y 5 3x 1 8 2 12x
y 5 4x 1 5 1 2x
y 5 3x 1 8 2 12x
y 5 6x 1 5
y 5 29x 1 8
slope 5 6
slope 5 29
3. 2x 1 3y 5 21
4. 8y 2 2x 5 24
2x 1 3y 5 21
8y 2 2x 5 24
3y 5 22x 1 21
8y 5 2x 1 24
__
__
2 y 5 2 x 1 7 3
y 5 2 x 1 3 8
2 slope 5 2 3
y 5 1 x 1 3 4
__
__
__
slope 5 1 4
Chapter 3 Skills Practice • 441
Lesson 3.4 Skills Practice
page 2
5. y 5 8
6. y 5 6x 2 2 1 x
y58
y 5 6x 2 2 1 x
y 5 0x 1 8
y 5 7x 2 2
slope 5 0
slope 5 7
7. y 5 25(2x 2 3)
8. 10y 2 6x 5 290
y 5 25(2x 2 3)
10y 2 6x 5 290
y 5 210x 1 15
10y 5 6x 2 90
slope 5 210
___ y 5 __ 3 x 2 9
y 5 6 x 2 9 10 5
__
slope 5 3 5 9. x 5 __ 1 2
10. 12y 2 4x 5 224
__
12y 2 4x 5 224 12y 5 4x 2 24
___ y 5 __ 1 x 2 2 3 slope 5 __ 1
y 5 4 x 2 2 12
3
442 • Chapter 3 Skills Practice
© 2011 Carnegie Learning
This equation represents a vertical line at x 5 1 . The slope is undefined. 2
Lesson 3.4 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ Graph each equation with your graphing calculator and sketch its graph on the given grid. Determine the slope of the line. 11. y 5 4x 1 2
y 16
slope 5 4
12 8 4 –8
–6
–4
–2
2
4
6
8
x
2
4
6
8
x
2
4
6
8
x
–4 –8 –12 –16
12. y 5 __ 1 x 1 1 2
y 8
__
slope 5 1 2
6 4 2 –8
–6
–4
–2 –2 –4 –6
© 2011 Carnegie Learning
–8
1 13. y 5 2__ x 2 5 3
y 8
__
1 slope 5 2 3
6 4 2 –8
–6
–4
–2 –2 –4 –6 –8
Chapter 3 Skills Practice • 443
Lesson 3.4 Skills Practice
page 4
14. y 5 22x 2 3
y 8
slope 5 22
6 4 2 –8
–6
–4
–2
2
4
6
8
x
2
4
6
8
x
2
4
6
8
x
–2 –4 –6 –8
15. y 5 __ 3 x 2 7 4
y 8
__
slope 5 3 4
6 4 2 –8
–6
–4
–2 –2 –4 –6 –8
2 16. y 5 2__ x 1 8 3
y 8
__
2 slope 5 2 3
4 2 –8
–6
–4
–2 –2 –4 –6 –8
444 • Chapter 3 Skills Practice
© 2011 Carnegie Learning
6
Lesson 3.5 Skills Practice Name_________________________________________________________ Date__________________________
Where it Crosses Determining y-Intercepts from Various Representations Vocabulary Define each term in your own words. 1. y-intercept the y-coordinate of the point where a graph crosses the y-axis, written in the form (0, y). 2. direct variation the relationship between two quantities x and y having a constant ratio such that when one increases or decreases by a specific amount the other value increases or decreases by a constant amount.
Problem Set Examine each linear graph and determine the y-intercept. Write the y-intercept in coordinate form. Show your work. 1.
Since the graph is a line, it will keep
y
increasing by 2 for every point going
10
backward.
(3, 9)
9
© 2011 Carnegie Learning
8
The next points to the left would be
(4, 7)
7
(2, 11), then (1, 13), then (0, 15).
6 5
The y-intercept is (0, 15).
4 3 2 1 0
1
2
3
4
5
6
7
8
9
10
x
Chapter 3 Skills Practice • 445
Lesson 3.5 Skills Practice
2.
page 2
Since the graph is a line, it will keep
y 5
increasing by 1 for every 2 points going
(2, 5)
backward. 4
(4, 4)
The next point to the left would be (0, 6). 3
The y-intercept is (0, 6). 2
1
0
1
3.
2
3
4
5
x
Since the graph is a line, it will keep
y 10
decreasing by 2 for every point going backward.
9 8
(3, 8)
The next points to the left would be (1, 4),
7 6
and then (0, 2).
(2, 6)
5
The y-intercept is (0, 2).
4 3 2 1
4.
1
2
3
4
5
6
7
8
9
10
x
Since the graph is a line, it will keep
y 20
decreasing by 12 for every 2 points
18
going backward.
16 14
The next point to the left would be (0, 210).
(4, 14)
12
The y-intercept is (0, 210).
10 8 6 4 2 0
(2, 2) 2
4
6
8
10
12
14
16
446 • Chapter 3 Skills Practice
18
20
x
© 2011 Carnegie Learning
0
Lesson 3.5 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ 5.
Since the graph is a line, it will keep
y 10
increasing by 5 for every point going
(3, 10)
backward.
9 8 7
The next points to the left would be
6
(2, 15), then (1, 20), then (0, 25).
5
(4, 5)
The y-intercept is (0, 25).
4 3 2 1 0
6.
1
2
3
4
5
6
7
8
9
x
10
Since the graph is a line, it will keep
y 5
decreasing by 3 for every point going
(3, 5)
backward. 4
The next points to the left would be (1, 21), and then (0, 24).
3
© 2011 Carnegie Learning
2
The y-intercept is (0, 24).
(2, 2)
1
0
1
2
3
4
5
x
Chapter 3 Skills Practice • 447
Lesson 3.5 Skills Practice
page 4
Determine the y-intercept for the linear relation represented in each table. Write the y-intercept as a coordinate pair.
8.
Counting backward in the table the x-values would
x
y
6
25
9
34
would be 16, 7.
12
43
The y-intercept is (0, 7).
15
52
x
y
4
12
6
13
would be 11, 10.
8
14
The y-intercept is (0, 10).
10
15
x
y
10
238
15
258
would be 218, 2.
20
278
The y-intercept is (0, 2).
25
298
9.
be 3, 0. Skip counting backward in the table the y-values
Counting backward in the table the x-values would be 2, 0. Skip counting backward in the table the y-values
Counting backward in the table the x-values would
448 • Chapter 3 Skills Practice
be 5, 0. Skip counting backward in the table the y-values
© 2011 Carnegie Learning
7.
Lesson 3.5 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ 10.
11.
Counting backward in the table the x-values would x
y
8
29
12
212
would be 26, 23.
16
215
The y-intercept is (0, 23).
20
218
x
y
6
35
9
53
would be 17, 21.
12
71
The y-intercept is (0, 21).
15
89
x
y
10
295
15
2145
would be 245, 5.
20
2195
The y-intercept is (0, 5).
25
2245
© 2011 Carnegie Learning
12.
be 4, 0. Skip counting backward in the table the y-values
Counting backward in the table the x-values would be 3, 0. Skip counting backward in the table the y-values
Counting backward in the table the x-values would be 5, 0. Skip counting backward in the table the y-values
Chapter 3 Skills Practice • 449
Lesson 3.5 Skills Practice
page 6
Determine the y-intercept for the linear relation represented in each equation. Write the y-intercept as a coordinate pair. 14. 5x 2 4y 5 120
3x 1 7y 5 63
5x 2 4y 5 120
3(0) 1 7y 5 63
5(0) 2 4y 5 120
7y 5 63
24y 5 120
y59
y 5 230
The y-intercept is (0, 9).
15. 2x 2 8y 5 248
The y-intercept is (0, 230).
16. 20x 1 15y 5 105
2x 2 8y 5 248
20x 1 15y 5 105
2(0) 2 8y 5 248
20(0) 1 15y 5 105
28y 5 248
15y 5 105
y56
y57
The y-intercept is (0, 6).
17. 216x 1 23y 5 253
The y-intercept is (0, 7).
18. 19x 1 17y 5 251
216x 1 23y 5 253
19x 1 17y 5 251
216(0) 1 23y 5 253
19(0) 1 17y 5 251
23y 5 253
17y 5 251
y 5 11
y 5 23
The y-intercept is (0, 23).
The y-intercept is (0, 11).
450 • Chapter 3 Skills Practice
© 2011 Carnegie Learning
13. 3x 1 7y 5 63
Lesson 3.5 Skills Practice
page 7
Name_________________________________________________________ Date__________________________ Determine the y-intercept for the linear relation represented in each context. Write the y-intercept as a coordinate pair. Explain what the y-intercept represents in the problem situation. 19. Carmen estimated that she would pay $19 to park in a downtown parking garage for a 3-hour event. After spending 5 hours downtown she paid $25 for parking. The difference between the prices is $6. The difference between the times parked is 2 hours. The rate paid is $3 per hour. The estimate for 3 hours of parking is $19, and $9 is due to the per hour rate. So, $10 must be an initial flat fee. The y-intercept is (0, 10).
20. Pedro is traveling on a toll road. He plans to exit the road 5 miles ahead at First Avenue and pay $1.75. He changes his plans and travels 9 miles to Butler Street and pays $2.75. The difference in the tolls is $1.00. The difference in the miles traveled is 4 miles. The rate paid is $0.25 per mile. The toll for traveling 5 miles is $1.75, and $1.25 is due to the per mile rate. So, $0.50 must be an initial flat fee. The y-intercept is (0, 0.50).
© 2011 Carnegie Learning
21. Alberto is saving for a new video game. After adding 2 weeks of his allowance to a savings account he has $105. After adding 3 more weeks of his allowance to his savings he has $150. Three weeks of allowance is $45 ($150 2 $105). Alberto’s allowance each week is $15. Counting backward by $15 per week from $105 for the first two weeks of allowance added is $75. The y-intercept is (0, 75). $75 is the initial balance of Alberto’s savings account before adding his allowance.
Chapter 3 Skills Practice • 451
Lesson 3.5 Skills Practice
page 8
22. Noah is renewing a magazine subscription. One package offers to renew the magazine for 3 years for $26. A second package offers to renew the magazine for 5 years for $38. The difference in the cost is $12. The difference in the length of the subscription is 2 years. The rate paid is $6 per year. The offer to renew for 3 years is $26, and $18 is due to the per year rate. So, $8 must be an initial flat fee. The y-intercept is (0, 8).
23. Serena received a gift card to the local movie theater. After going to 2 movies, the balance of her gift card dropped to $64. After going to 3 more movies, the balance of her gift card dropped to $40. Three movies cost $24 ($64 2 $40). The cost is $8 per movie. Counting forward by $8 per movie from $64 for the first two movies is $80. The y-intercept is (0, 80). $80 is the initial value
© 2011 Carnegie Learning
of Serena’s gift card.
452 • Chapter 3 Skills Practice
Lesson 3.6 Skills Practice Name_________________________________________________________ Date__________________________
Slope-Intercept Form Determining the Rate of Change and y-Intercept
Vocabulary Match each definition to its corresponding term. a. point-slope form
1. Ax 1 By 5 C, where A, B, and C are constants and A and B are not both zero. b. standard form 2. m(x 2 x1) 5 (y 2 y1), a linear equation that passes
b. standard form
through the point (x1, y1) and has slope m a. point-slope form
Problem Set Sketch the graph of each line. 2 x 1 5 2. y 5 2 __ 3
1. y 5 3x 1 2
y
© 2011 Carnegie Learning
y
x
x
Chapter 3 Skills Practice • 453
Lesson 3.6 Skills Practice
3. y 5 __ 1 x 2 6 2
page 2
4. y 5 24x 2 3 y
y
x
3 x 2 4 5. y 5 2 __ 4
3 x 1 1 6. y 5 __ 8 y
y
x
x
Determine the y-intercept of each line given the slope and a point that lies on the line. 7. m 5 4 and (2, 13) y 5 mx 1 b 13 5 4(2) 1 b 13 5 8 1 b 55b
454 • Chapter 3 Skills Practice
8. m 5 __ 3 and (4, 26) 2 y 5 mx 1 b
__
26 5 3 (4) 1 b 2 26 5 6 1 b 212 5 b
© 2011 Carnegie Learning
x
Lesson 3.6 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ 2 and (5, 12) 9. m 5 2 __ 5 y 5 mx 1 b
y 5 mx 1 b
__
__ 3 1 b 8 5 2 __ 4 3 5 b 8 __
12 5 2 2 (5) 1 b 5
8 5 2 1 (3) 1 b 4
12 5 22 1 b 14 5 b
4
(
)
1 11. m 5 7 and 2, 13 __ 2 y 5 mx 1 b
__ 1 5 14 1 b 13 __ 2 1 5 b 2 __
13 1 5 7(2) 1 b 2
12. m 5 __ 3 and (8, 21) 4 y 5 mx 1 b
__
21 5 3 (8) 1 b 4 21 5 6 1 b 15 5 b
© 2011 Carnegie Learning
2
1 and (3, 8) 10. m 5 2 __ 4
Chapter 3 Skills Practice • 455
Lesson 3.6 Skills Practice
page 4
Write the equation of each line given two points that lie on the line. 13. (3, 25) and (4, 31) Calculate the slope, m.
Calculate the y-intercept, b.
y 2y m 5 _______ x2 2 x1
y 5 mx 1 b
5 ________ 31 2 25 423
25 5 6(3) 1 b
5 6 1
25 5 18 1 b
Substitute m and b.
75b
2
1
__
y 5 mx 1 b y 5 6x 1 7
14. (10, 22) and (15, 24)
y 2y m 5 _______ x2 2 x1 2
1
Calculate the y-intercept, b. y 5 mx 1 b
__
5 ________ 24 2 22 15 2 10
22 5 2 (10) 1 b 5
5 2 5
22 5 4 1 b
Substitute m and b.
18 5 b
__
y 5 mx 1 b
__
y 5 2 x 1 18 5
456 • Chapter 3 Skills Practice
© 2011 Carnegie Learning
Calculate the slope, m.
Lesson 3.6 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ 15. (6, 1) and (18, 21) Calculate the slope, m.
Calculate the y-intercept, b.
y 2y m 5 _______ x2 2 x1
y 5 mx 1 b
5 _______ 21 2 1 18 2 6
1 5 2 1 (6) 1 b 6
5 ___ 22 12
1 5 21 1 b
5 2 1 6
25b
2
1
__
__
Substitute m and b. y 5 mx 1 b
__
y 5 2 1 x 1 2 6
(
(
)
)
3 and 5, 39 __ 3 16. 2, 15 __ 4 4 Calculate the slope, m. y2 2 y1 m 5 _______ x 2x © 2011 Carnegie Learning
2
1
3 2 15 __ 3 39 __ 4 4 __________ 5 522
___
5 24 3
Calculate the y-intercept, b. y 5 mx 1 b
__ 3 5 16 1 b 15 __ 4 1 5 b 2 __
15 3 5 8(2) 1 b 4
4
5 8 Substitute m and b. y 5 mx 1 b
__
y 5 8x 2 1 4
Chapter 3 Skills Practice • 457
Lesson 3.6 Skills Practice
page 6
17. (7, 1) and (21, 11) Calculate the slope, m.
Calculate the y-intercept, b.
y 2y m 5 _______ x2 2 x1
y 5 mx 1 b
5 _______ 11 2 1 21 2 7
1 5 5 (7) 1 b 7
5 10 14
1551b
2
1
___ 5 5 __
__
24 5 b
7
Substitute m and b. y 5 mx 1 b
__
y 5 5 x 2 4 7
18. (2, 27) and (4, 210)
y 2y m 5 _______ x2 2 x1 2
1
Calculate the y-intercept, b. y 5 mx 1 b
__
210 2 (27) 5 ___________ 422
27 5 2 3 (2) 1 b 2
23 5 ___ 2
27 5 23 1 b
Substitute m and b.
24 5 b
y 5 mx 1 b
__
y 5 2 3 x 2 4 2
458 • Chapter 3 Skills Practice
© 2011 Carnegie Learning
Calculate the slope, m.
Lesson 3.6 Skills Practice
page 7
Name_________________________________________________________ Date__________________________ Write the equation of each line given the slope and a point that lies on the line. Write the equation in slope-intercept form. 19. slope 5 9 and (2, 21) m(x 2 x1) 5 (y 2 y1)
20. slope 5 22 and (4, 7) m(x 2 x1) 5 (y 2 y1)
9(x 2 2) 5 (y 2 21)
22(x 2 4) 5 (y 2 7)
9x 2 18 5 y 2 21
22x 1 8 5 y 2 7
9x 1 3 5 y
22x 1 15 5 y
y 5 9x 1 3
21. slope 5 __ 2 and (10, 13) 5
© 2011 Carnegie Learning
m(x 2 x1) 5 (y 2 y1)
__ 2 (x 2 10) 5 (y 2 13) 5 2 x 2 4 5 y 2 13 __ 5
__
2 x 1 9 5 y 5
__
y 5 2 x 1 9 5
y 5 22x 1 15
4 and (6, 216) 22. slope 5 2 __ 3 m(x 2 x1) 5 (y 2 y1)
__
2 4 (x 2 6) 5 (y 2 (216)) 3
__
2 4 x 1 8 5 y 1 16 3
__
2 4 x 2 8 5 y 3
__
y 5 2 4 x 2 8 3
Chapter 3 Skills Practice • 459
Lesson 3.6 Skills Practice
23. slope 5 __ 1 and (9, 212) 3
page 8
5 and (24, 26) 24. slope 5 2 ___ 12 m(x 2 x1) 5 (y 2 y1)
m(x 2 x1) 5 (y 2 y1)
__ 1 (x 2 9) 5 (y 2 (212)) 3
__ 1 x 2 3 5 y 1 12 3
__
1 x 2 15 5 y 3
___ 5 x 1 10 5 y 1 6 2 ___
2 5 (x 2 24) 5 (y 2 (26)) 12 12
___
2 5 x 1 4 5 y 12
__
___
y 5 2 5 x 1 4 12
y 5 1 x 2 15 3
Calculate the y-intercept and x-intercept for each linear equation in standard form. Sketch the graph of the line. 25. 4x 1 6y 5 48
y 16
4x 1 6y 5 48
12 8
4(0) 1 6y 5 48
4
6y 5 48
–16 –12 –8
–4
4
8
12
16 x
–4
y58
–8 –12
4x 1 6y 5 48 4x 1 6(0) 5 48 4x 5 48 x 5 12 The x-intercept is (12, 0).
460 • Chapter 3 Skills Practice
–16
© 2011 Carnegie Learning
The y-intercept is (0, 8).
Lesson 3.6 Skills Practice
page 9
Name_________________________________________________________ Date__________________________ 26. 3x 1 15y 5 135
y 8
3x 1 15y 5 135
6 4
3(0) 1 15y 5 135
2
15y 5 135
–40 –30 –20 –10
10
20
30
40 x
–2
y59
–4 –6
The y-intercept is (0, 9).
–8
3x 1 15y 5 135 3x 1 15(0) 5 135 3x 5 135 x 5 45
© 2011 Carnegie Learning
The x-intercept is (45, 0).
Chapter 3 Skills Practice • 461
Lesson 3.6 Skills Practice
27. 22x 1 8y 5 56
page 10
y 8
22x 1 8y 5 56
6 4
22(0) 1 8y 5 56
2
8y 5 56
–32 –24 –16 –8
8
16
24
32 x
–2
y57
–4 –6
The y-intercept is (0, 7).
–8
22x 1 8y 5 56 22x 1 8(0) 5 56 22x 5 56 x 5 228
© 2011 Carnegie Learning
The x-intercept is (228, 0).
462 • Chapter 3 Skills Practice
Lesson 3.6 Skills Practice
page 11
Name_________________________________________________________ Date__________________________ 28. 5x 1 6y 5 290
y 16
5x 1 6y 5 290
12 8
5(0) 1 6y 5 290
4
6y 5 290
–16 –12 –8
–4
4
8
12
16 x
–4
y 5 215
–8 –12
The y-intercept is (0, 215).
–16
5x 1 6y 5 290 5x 1 6(0) 5 290 5x 5 290 x 5 218
© 2011 Carnegie Learning
The x-intercept is (218, 0).
Chapter 3 Skills Practice • 463
Lesson 3.6 Skills Practice
page 12
29. 2x 1 8y 5 224
y 16
2x 1 8y 5 224
12 8
2(0) 1 8y 5 224
4
8y 5 224
–16 –12 –8
–4
4
8
12
16 x
–4
y 5 23
–8 –12
The y-intercept is (0, 23).
–16
2x 1 8y 5 224 2x 1 8(0) 5 224 2x 5 224 x 5 212
© 2011 Carnegie Learning
The x-intercept is (212, 0).
464 • Chapter 3 Skills Practice
Lesson 3.6 Skills Practice
page 13
Name_________________________________________________________ Date__________________________ 30. 7x 2 3y 5 242
y 16
7x 2 3y 5 242
12 8
7(0) 2 3y 5 242
4
23y 5 242
–16 –12 –8
–4
4
8
12
16 x
–4
y 5 14
–8 –12
The y-intercept is (0, 14).
–16
7x 2 3y 5 242 7x 2 3(0) 5 242 7x 5 242 x 5 26
© 2011 Carnegie Learning
The x-intercept is (26, 0).
Chapter 3 Skills Practice • 465
© 2011 Carnegie Learning
466 • Chapter 3 Skills Practice
