name date · 540 600 length of track remaining in bobsled run 90 a b c point c is at (15, 150)....

Chapter 3 Skills Practice • 409

© 2

011

Car

negi

e Le

arni

ng

Hitting the SlopesDetermining Rate of Change from a Graph

VocabularyMatch each definition to its corresponding term.

1. rate which has a second term of 1 unit

e. unit rate

a. rate of change

2. ratio in which the units of the quantities being compared are different

d. rate

b. per

3. the vertical change from one point to another point on a graph

g. rise

c. run

4. phrase used when a rate is used to describe a rate of increase

(or decrease) in a real-life situation

a. rate of change

d. rate

5. the rate of change on a graph written as the vertical change from one

point to another over the horizontal change of the same two points

f. rise ____ run

e. unit rate

6. means “for each” or “for every”

b. per

f. rise ____ run

7. the horizontal change from one point to another point on a graph

c. run

g. rise

Lesson 3.1 Skills Practice

Name ________________________________________________________ Date _________________________

410 • Chapter 3 Skills Practice

© 2

011

Car

negi

e Le

arni

ng

Lesson 3.1 Skills Practice page 2

Problem SetWrite 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.

Time (in seconds)

Trac

k le

ngth

(in

met

ers)

x86

180

10 12 14 16 18 20420

y

270

360

450

540

600

Length of Track Remaining in Bobsled Run

90

A

B

C

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

.


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

2. Use the graph to write a rate that compares the track length to the change in time at point B.

Time (in seconds)

Trac

k le

ngth

(in

met

ers)

x86

180

10 12 14 16 18 20420

y

270

360

450

540

600


90

A

B

C

Point B is at (10, 300).



.


© 2

011

Car

negi

e Le

arni

ng


3. Use the graph to write a rate that compares the track length to the change in time at point C.

Time (in seconds)

Trac

k le

ngth

(in

met

ers)

x86

180

10 12 14 16 18 20420

y

270

360

450

540

600


90

A

B

C

Point C is at (15, 150).



.


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

4. Use the graph to write a rate that compares the change in elevation to the change in time

at point D.

Time (in minutes)

Dis

tanc

e (in

feet

)

x86 10 12 14 16 18 20420

y Distance Ascended by a Mountain Climber

20

40

60

80

100

120

140

160

180

200

D

E

F

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 .


© 2

011

Car

negi

e Le

arni

ng



at point E.

Time (in minutes)

Dis

tanc

e (in

feet

)

x86 10 12 14 16 18 20420


20

40

60

80

100

120

140

160

180

200

D

E

F

Point E is at (12, 120).


in time is 12 minutes. So, the rate of increase is 120 ____ 12

.


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________


at point F.

Time (in minutes)

Dis

tanc

e (in

feet

)

x86 10 12 14 16 18 20420


20

40

60

80

100

120

140

160

180

200

D

E

F

Point F is at (18, 180).


in time is 18 minutes. So, the rate of increase is 180 ____ 18

.


© 2

011

Car

negi

e Le

arni

ng


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.

Time (in seconds)

Ele

va

tio

n (

in m

ete

rs)

x86 10 12 14 16 18 20420

y

20

40

60

80

100

120

140

160

180

200

B

A

Sample answer:

Point A: (4, 90)

Point B: (6, 50)

rise ____ run 5 240 meters ___________ 2 seconds

unit rate 5 220 meters ___________ 1 second

8.

Time (in seconds)

Ele

va

tio

n (

in m

ete

rs)

x43 5 6 7 8 9 10210

y

5

10

15

20

25

30

35

40

45

50

B

A

Sample answer:

Point A: (2, 35)

Point B: (4, 25)


unit rate 5 25 meters __________ 1 second


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

9.

Time (in minutes)

Dis

tan

ce

(in

fe

et)

x86 10 12 14 16 18 20420

y

6

12

18

24

30

36

42

48

54

60

A

B

Sample answer:

Point A: (4, 24)

Point B: (6, 36)

rise ____ run 5 12 feet __________ 2 minutes

unit rate 5 6 feet _________ 1 minute

10.

Time (in hours)

Dis

tan

ce

(in

mile

s)

x43 5 6 7 8 9 10210

y

200

150

100

50

A

B

Sample answer:

Point A: (6, 90)

Point B: (8, 120)

rise ____ run 5 30 miles ________ 2 hours

unit rate 5 15 miles ________ 1 hour


© 2

011

Car

negi

e Le

arni

ng


11.

Time (in seconds)

Ele

va

tio

n (

in m

ete

rs)

x86 10 12 14 16 18 20420

y

6

12

18

24

30

36

42

48

54

60

B

A

Sample answer:

Point A: (2, 36)

Point B: (4, 12)



12.

Time (in hours)D

ista

nce

(in

mile

s)

x43 5210

y

30

60

90

120

150

180

210

240

270

300B

A

Sample answer:

Point A: (2, 150)

Point B: (4, 300)

rise ____ run 5 150 miles _________ 2 hours

unit rate 5 75 miles ________ 1 hour


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

Determine the unit rate of change for lines A and B on each graph.

13.

Number of items

Cos

t (in

dol

lars

)

x86 1042 75 9310

y

1

2

3

4

5

6

7

8

9

10

C

D

F

E

AB

Line A sample answer:

Point C: (0, 0)

Point D: (1, 1.5)

rise ____ run 5 1.5 dollars __________ 1 item

unit rate 5 1.5 dollars __________ 1 item

Line B sample answer:

Point E: (0, 0)

Point F: (1, 3)

rise ____ run 5 3 dollars _________ 1 item

unit rate 5 3 dollars _________ 1 item


© 2

011

Car

negi

e Le

arni

ng


14.

Number of items

Cos

t (in

dol

lars

)

x86 1042 75 9310

y

2

4

6

8

10

12

14

16

18

20

E

F

AB

C

D


Point C: (1, 7)

Point D: (2, 9)




Point E: (1, 9)

Point F: (2, 13)




© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

15.

Time (in seconds)

Ele

vatio

n (in

met

ers)

x1612 2084 1410 18620

y

10

20

30

40

E

AB

F

D

C


Point C: (2, 30)

Point D: (4, 20)




Point E: (2, 24)

Point F: (4, 8)




© 2

011

Car

negi

e Le

arni

ng


16.

Time (in seconds)

Ele

vatio

n (in

met

ers)

x86 1042 75 9310

y

10

20

30

40

50

60

70

80

90

100

E

AB

F

D

C


Point C: (1, 70)

Point D: (3, 50)




Point E: (1, 60)

Point F: (3, 20)




© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

17.

Number of items

Cos

t (in

dol

lars

)

x43 5210

y

10

20

30

40A

B

E

F

D

C


Point C: (0, 0)

Point D: (2, 18)

rise ____ run 5 18 dollars __________ 2 items



Point E: (0, 0)

Point F: (2, 6)

rise ____ run 5 6 dollars _________ 2 items



© 2

011

Car

negi

e Le

arni

ng


18.

Time (in minutes)

Ele

vatio

n (in

feet

)

x202 4 6 8 10 12 14 16 180

y

2

4

6

8

10

12

14

16

18

20

B

AF

E

C

D


Point C: (4, 16)

Point D: (8, 14)


unit rate 5 20.5 feet _________ 1 minute


Point E: (2, 13)

Point F: (4, 8)


unit rate 5 22.5 feet _________ 1 minute


© 2

011

Car

negi

e Le

arni

ng

At the ArcadeDetermining Rate of Change from a Table

VocabularyDefine 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 SetUse 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.

1.

Number of Balloons

Total Cost of Balloons

(in Dollars)

2 6

4 12

6 18

8 24

Sample answer:

Number of balloons 5 4 2 2

5 2

Cost of balloons 5 12 2 6

5 6

Rate of change 5 6 dollars __________ 2 balloons

Unit rate 5 3 dollars _________ 1 balloon


Name ________________________________________________________ Date _________________________


© 2

011

Car

negi

e Le

arni

ng


2.Number of

LawnsTotal Earned (in Dollars)

3 25.50

5 42.50

7 59.50

9 76.50

Sample answer:

Number of lawns 5 5 2 3

5 2

Total earned 5 42.50 2 25.50

5 17

Rate of change 5 17 dollars __________ 2 lawns

Unit rate 5 8.5 dollars __________ 1 lawn

3.Number of

TouchdownsTotal 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

5 6

Rate of change 5 6 points _____________

1 touchdown

Unit rate 5 6 points _____________

1 touchdown


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

4.Number of

Minutes on an Exercise Bike

Total Number of Calories Burned

15 180

30 360

45 540

60 720

Sample answer:

Number of minutes 5 30 2 15

5 15

Number of calories 5 360 2 180

5 180

Rate of change 5 180 calories ____________ 15 minutes

Unit rate 5 12 calories ___________ 1 minute

5.

Number of HoursTotal Number

of Miles Traveled

2 130

5 325

8 520

11 715

Sample answer:

Number of hours 5 5 2 2

5 3

Number of miles 5 325 2 130

5 195

Rate of change 5 195 miles _________ 3 hours

Unit rate 5 65 miles ________ 1 hour


© 2

011

Car

negi

e Le

arni

ng


6.Number of Songs

Downloaded

Total Cost of Songs

(in Dollars)

10 9.90

20 19.80

30 29.70

40 39.60

Sample answer:

Number of songs 5 20 2 10

5 10

Cost of songs 5 19.80 2 9.90

5 9.9

Rate of change 5 $9.90 _________ 10 songs

Unit rate 5 $0.99 _______ 1 song

Use the formula y2 2 y1 _______ x2 2 x1

to calculate the unit rate of change for the data in each table. The rate of

change is constant for the data in each table.

7.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)

y2 2 y1 ________

x2 2 x1

5 2 2 1 ______ 4 2 2

5 1 __ 2

5 0.5 ___ 1

Unit rate 5 $0.50 _______ 1 ticket


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

8.x y

22 8

0 0

2 28

4 216

Sample answer:

( x1, y1 ) 5 (22, 8)

( x2, y2 ) 5 (2, 28)

y2 2 y1 ________

x2 2 x1

5 28 2 8 ________ 2 2 (22)

5 216 _____ 4

5 24 ___ 1

Unit rate 5 24 ___ 1

9.

Number of Photos Printed

Total Cost of Photos

(in Dollars)

10 2

20 4

30 6

40 8

Sample answer:

( x1, y1 ) 5 (10, 2)

( x2, y2 ) 5 (20, 4)

y2 2 y1 ________

x2 2 x1

5 4 2 2 ________ 20 2 10

5 2 ___ 10

5 0.2 ___ 1

Unit rate 5 $0.20 ________ 1 photo


© 2

011

Car

negi

e Le

arni

ng


10.x y

3 27

5 45

7 63

9 81

Sample answer:

( x1, y1 ) 5 (3, 27)

( x2, y2 ) 5 (5, 45)

y2 2 y1 ________

x2 2 x1

5 45 2 27 ________ 5 2 3

5 18 ___ 2

5 9 __ 1

Unit rate 5 9 __ 1

11.

Number of Greeting Cards

Total Cost of Greeting Cards

(in Dollars)

2 6.50

3 9.75

6 19.50

8 26.00

Sample answer:

( x1, y1 ) 5 (2, 6.50)

( x2, y2 ) 5 (3, 9.75)

y2 2 y1 ________

x2 2 x1

5 9.75 2 6.50 ___________ 3 2 2

5 3.25 _____ 1

Unit rate 5 $3.25 ______ 1 card


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

12.x y

23 54

1 218

3 254

7 2126

Sample answer:

( x1, y1 ) 5 (23, 54)

( x2, y2 ) 5 (1, 218)

y2 2 y1 ________

x2 2 x1

5 218 2 54 __________ 1 2 (23)

5 272 _____ 4

5 218 _____ 1

Unit rate 5 218 _____ 1

Calculate the rate of change between the points listed in each table. Determine if the table represents a

linear relationship.

13.x y

2 14

5 35

7 49

10 70

Yes, the table represents a linear

relationship.

(2, 14) and (5, 35) (5, 35) and (7, 49)

35 2 14 ________ 5 2 2

5 21 ___ 3

49 2 35 ________ 7 2 5

5 14 ___ 2

5 7 __ 1

5 7 __ 1

(7, 49) and (10, 70)

70 2 49 ________ 10 2 7

5 21 ___ 3

5 7 __ 1


© 2

011

Car

negi

e Le

arni

ng


14.x y

210 50

22 10

4 220

14 270


relationship.

(210, 50) and (22, 10) (22, 10) and (4, 220)

10 2 50 ___________ 22 2 (210)

5 240 _____ 8

220 2 10 __________ 4 2 (22)

5 230 _____ 6

5 25 ___ 1

5 25 ___ 1

(4, 220) and (14, 270)

270 2 (220)

____________ 14 2 4

5 250 _____ 10

5 25 ___ 1

15.x y

21 224

2 48

4 90

8 192

No, the table does not represent a linear

relationship.

(21, 224) and (2, 48) (2, 48) and (4, 90)

48 2 (224)

___________ 2 2 (21)

5 72 ___ 3

90 2 48 ________ 4 2 2

5 42 ___ 2

5 24 ___ 1

5 21 ___ 1

(4, 90) and (8, 192)

192 2 90 _________ 8 2 4

5 102 ____ 4

5 25.5 _____ 1


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

16.x y

26 12

23 6

3 26

6 210


relationship.

(26, 12) and (23, 6) (23, 6) and (3, 26)

6 2 12 __________ 23 2 (26)

5 26 ___ 3

26 2 6 ________ 3 2 (23)

5 212 _____ 6

5 22 ___ 1

5 22 ___ 1

(3, 26) and (6, 210)

210 2 (26)

___________ 6 2 3

5 24 ___ 3

17.x y

2 13.5

5 33.75

10 67.5

15 101.25


relationship.

(2, 13.5) and (5, 33.75)

33.75 2 13.5 ____________ 5 2 2

5 20.25 ______ 3

5 6.75 _____ 1

(5, 33.75) and (10, 67.5)

67.5 2 33.75 ____________ 10 2 5

5 33.75 ______ 5

5 6.75 _____ 1

(10, 67.5) and (15, 101.25)

101.25 2 67.5 _____________ 15 2 10

5 33.75 ______ 5

5 6.75 _____ 1


© 2

011

Car

negi

e Le

arni

ng


18.x y

24 238

21 29.5

2 19

3 27


relationship.

(24, 238) and (21, 29.5)

29.5 2 (238)

_____________ 21 2 (24)

5 28.5 _____ 3

5 9.5 ___ 1

(21, 29.5) and (2, 19)

19 2 (29.5)

___________ 2 2 (21)

5 28.5 _____ 3

5 9.5 ___ 1

(2, 19) and (3, 27)

27 2 19 ________ 3 22

5 8 __ 1


© 2

011

Car

negi

e Le

arni

ng

To Put It in ContextDetermining Rate of Change from a Context

Problem SetDetermine 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

2.5 hours ____________ 10 bracelets

5 0.25 hour __________ 1 bracelet

The unit rate is 0.25 hour __________ 1 bracelet

or 15 minutes per 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

12,000 people

______________ 4 days

5 3000 people

____________ 1 day

The unit rate is 3000 people

____________ 1 day

.


Name ________________________________________________________ Date _________________________


© 2

011

Car

negi

e Le

arni

ng


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

$1.50 __________ 2 toppings

5 $0.75 _________ 1 topping

The unit rate is $0.75 _________ 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?

5 batches __________ 2.5 hours

5 2 batches __________ 1 hour

The unit rate is 2 batches __________ 1 hour

.

5. Nelson is selling his photographs at an art festival. The festival is open for 6 hours each day for

3 days. At the conclusion of the festival, Nelson has sold 54 photographs. What is the unit rate of

the number of photographs sold per hour?

3 days 3 6 hours per day 5 18 hours total

54 photographs

_______________ 18 hours

5 3 photographs

______________ 1 hour

The unit rate is 3 photographs

______________ 1 hour

.


© 2

011

Car

negi

e Le

arni

ng


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?

Package 5 $18.75 __________ 25 tickets

5 $0.75 ________ 1 ticket

The unit rate is $0.75 _______ 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

$680 _______ 85 pies

5 $8 ______ 1 pie

The unit rate is $8 ______ 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.

7.5 miles _________ 5 hours

5 1.5 miles _________ 1 hour

The unit rate is 1.5 miles _________ 1 hour

.


© 2

011

Car

negi

e Le

arni

ng


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

$29.98 _________ 2 T-shirts

5 $14.99 ________ 1 T-shirt

The unit rate is $14.99 ________ 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

75 minutes ___________ 150 photos

5 0.5 minute __________ 1 photo

The unit rate is 0.5 minute __________ 1 photo

.


© 2

011

Car

negi

e Le

arni

ng


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

the information to coordinate points and use the formula y2 2 y1 _______ x2 2 x1

to answer the question.

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)

y2 2 y1 _______ x2 2 x1

5 715 2 390 __________ 11 2 6

5 325 miles _________ 5 hours

5 65 miles ________ 1 hour

The unit rate is 65 miles ________ 1 hour

.


© 2

011

Car

negi

e Le

arni

ng


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

information to coordinate points and use the formula y2 2 y1 _______ x2 2 x1

to answer the question.

(x1, y1) 5 (7, 30)

(x2, y2) 5 (15, 90)

y2 2 y1 _______ x2 2 x1

5 90 2 30 ________ 15 2 7

5 60 days

________ 8 books

5 7.5 days

________ 1 book

The unit rate is 7.5 days

________ 1 book

.


© 2

011

Car

negi

e Le

arni

ng

All Together Now!Determining Rate of Change from an Equation

VocabularyGive 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 SetDetermine the slope of the line represented by each equation.


Name ________________________________________________________ Date _________________________

1. y 5 4x 1 5 1 2x

y 5 4x 1 5 1 2x

y 5 6x 1 5

slope 5 6

3. 2x 1 3y 5 21

2x 1 3y 5 21

3y 5 22x 1 21

y 5 22 __ 3 x 1 7

slope 5 22 __ 3

2. y 5 3x 1 8 2 12x

y 5 3x 1 8 2 12x

y 5 29x 1 8

slope 5 29

4. 8y 2 2x 5 24

8y 2 2x 5 24

8y 5 2x 1 24

y 5 2 __ 8

x 1 3

y 5 1 __ 4

x 1 3

slope 5 1 __ 4


© 2

011

Car

negi

e Le

arni

ng


5. y 5 8

y 5 8

y 5 0x 1 8

slope 5 0

7. y 5 25(2x 2 3)

y 5 25(2x 2 3)

y 5 210x 1 15

slope 5 210

9. x 5 1 __ 2

This equation represents a vertical line

at x 5 1 __ 2

. The slope is undefined.

6. y 5 6x 2 2 1 x

y 5 6x 2 2 1 x

y 5 7x 2 2

slope 5 7

8. 10y 2 6x 5 290

10y 2 6x 5 290

10y 5 6x 2 90

y 5 6 ___ 10

x 2 9

y 5 3 __ 5

x 2 9

slope 5 3 __ 5

10. 12y 2 4x 5 224

12y 2 4x 5 224

12y 5 4x 2 24

y 5 4 ___ 12

x 2 2

y 5 1 __ 3

x 2 2

slope 5 1 __ 3


© 2

011

Car

negi

e Le

arni

ng


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

slope 5 4

x

y

–16

–12

–8

–4

4

8

12

16

–2–4 2 4 6–6–8 8

12. y 5 1 __ 2

x 1 1

slope 5 1 __ 2

x

y

–8

–6

–4

–2

2

4

6

8

–2–4 2 4 6–6–8 8

13. y 5 21 __ 3 x 2 5

slope 5 21 __ 3

x

y

–8

–6

–4

–2

2

4

6

8

–2–4 2 4 6–6–8 8


© 2

011

Car

negi

e Le

arni

ng


14. y 5 22x 2 3

slope 5 22

x

y

–8

–6

–4

–2

2

4

6

8

–2–4 2 4 6–6–8 8

15. y 5 3 __ 4

x 2 7

slope 5 3 __ 4

x

y

–8

–6

–4

–2

2

4

6

8

–2–4 2 4 6–6–8 8

16. y 5 22 __ 3 x 1 8

slope 5 22 __ 3

x

y

–8

–6

–4

–2

2

4

6

8

–2–4 2 4 6–6–8 8


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

Where it CrossesDetermining y-Intercepts from Various Representations

VocabularyDefine 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 SetExamine each linear graph and determine the y-intercept. Write the y-intercept in coordinate form.

Show your work.

1.

x109876543210

y

10

9

8

7

6

5

4

3

2

1

(3, 9)

(4, 7)

Since the graph is a line, it will keep

increasing by 2 for every point going

backward.

The next points to the left would be

(2, 11), then (1, 13), then (0, 15).

The y-intercept is (0, 15).


© 2

011

Car

negi

e Le

arni

ng


2.

x543210

y

5

4

3

2

1

(2, 5)

(4, 4)


increasing by 1 for every 2 points going

backward.

The next point to the left would be (0, 6).


3.

x100

y

10

10987654321

10

9

8

7

6

5

4

3

2

1

(3, 8)

(2, 6)


decreasing by 2 for every point going

backward.

The next points to the left would be (1, 4),

and then (0, 2).


4.

x20181614121086420

y

20

18

16

14

12

10

8

6

4

2

(4, 14)

(2, 2)


decreasing by 12 for every 2 points

going backward.

The next point to the left would be (0, 210).



© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

5.

x100

y

10

987654321

9

8

7

6

5

4

3

2

1

(4, 5)

(3, 10)


increasing by 5 for every point going

backward.

The next points to the left would be

(2, 15), then (1, 20), then (0, 25).


6.

x543210

y

5

4

3

2

1

(3, 5)

(2, 2)


decreasing by 3 for every point going

backward.

The next points to the left would

be (1, 21), and then (0, 24).



© 2

011

Car

negi

e Le

arni

ng


Determine the y-intercept for the linear relation represented in each table. Write the y-intercept as a

coordinate pair.

7.x y

6 25

9 34

12 43

15 52

Counting backward in the table the x-values would

be 3, 0.

Skip counting backward in the table the y-values

would be 16, 7.


8.x y

4 12

6 13

8 14

10 15


be 2, 0.


would be 11, 10.


9. x y

10 238

15 258

20 278

25 298


be 5, 0.


would be 218, 2.



© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

10.x y

8 29

12 212

16 215

20 218


be 4, 0.


would be 26, 23.


11. x y

6 35

9 53

12 71

15 89


be 3, 0.


would be 17, 21.


12. x y

10 295

15 2145

20 2195

25 2245


be 5, 0.


would be 245, 5.



© 2

011

Car

negi

e Le

arni

ng


Determine the y-intercept for the linear relation represented in each equation. Write the

y-intercept as a coordinate pair.

13. 3x 1 7y 5 63

3x 1 7y 5 63

3(0) 1 7y 5 63

7y 5 63

y 5 9


14. 5x 2 4y 5 120

5x 2 4y 5 120

5(0) 2 4y 5 120

24y 5 120

y 5 230


15. 2x 2 8y 5 248

2x 2 8y 5 248

2(0) 2 8y 5 248

28y 5 248

y 5 6


16. 20x 1 15y 5 105

20x 1 15y 5 105

20(0) 1 15y 5 105

15y 5 105

y 5 7


17. 216x 1 23y 5 253

216x 1 23y 5 253

216(0) 1 23y 5 253

23y 5 253

y 5 11


18. 19x 1 17y 5 251

19x 1 17y 5 251

19(0) 1 17y 5 251

17y 5 251

y 5 23



© 2

011

Car

negi

e Le

arni

ng


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).

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.


© 2

011

Car

negi

e Le

arni

ng


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

of Serena’s gift card.


© 2

011

Car

negi

e Le

arni

ng

Slope-Intercept FormDetermining the Rate of Change and y-Intercept

VocabularyMatch each definition to its corresponding term.

1. Ax 1 By 5 C, where A, B, and C are constants

and A and B are not both zero.

b. standard form

a. point-slope form

2. m(x 2 x1) 5 (y 2 y1), a linear equation that passes

through the point (x1, y1) and has slope m

a. point-slope form

b. standard form

Problem SetSketch the graph of each line.

1. y 5 3x 1 2

y

x

2. y 5 2 2 __ 3

x 1 5

y

x


Name ________________________________________________________ Date _________________________


© 2

011

Car

negi

e Le

arni

ng


3. y 5 1 __ 2

x 2 6

y

x

4. y 524x 2 3

y

x

5. y 5 2 3 __ 4

x 2 4

y

x

6. y 5 3 __ 8

x 1 1

y

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

5 5 b

8. m 5 3 __ 2

and (4, 26)

y 5 mx 1 b

26 5 3 __ 2 (4) 1 b

26 5 6 1 b

212 5 b


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

9. m 5 2 2 __ 5

and (5, 12)

y 5 mx 1 b

12 5 2 2 __ 5

(5) 1 b

12 5 22 1 b

14 5 b

10. m 5 2 1 __ 4

and (3, 8)

y 5 mx 1 b

8 5 2 1 __ 4

(3) 1 b

8 5 2 3 __ 4

1 b

8 3 __ 4 5 b

11. m 5 7 and ( 2, 13 1 __ 2 )

y 5 mx 1 b

13 1 __ 2

5 7(2) 1 b

13 1 __ 2

5 14 1 b

2 1 __ 2

5 b

12. m 5 3 __ 4

and (8, 21)

y 5 mx 1 b

21 5 3 __ 4

(8) 1 b

21 5 6 1 b

15 5 b


© 2

011

Car

negi

e Le

arni

ng


Write the equation of each line given two points that lie on the line.

13. (3, 25) and (4, 31)

Calculate the slope, m.

m 5 y2 2 y1 _______ x2 2 x1

5 31 2 25 ________ 4 2 3

5 6 __ 1

Substitute m and b.

y 5 mx 1 b

y 5 6x 1 7

Calculate the y-intercept, b.

y 5 mx 1 b

25 5 6(3) 1 b

25 5 18 1 b

7 5 b

14. (10, 22) and (15, 24)


m 5 y2 2 y1 _______ x2 2 x1

5 24 2 22 ________ 15 2 10

5 2 __ 5

Substitute m and b.

y 5 mx 1 b

y 5 2 __ 5

x 1 18


y 5 mx 1 b

22 5 2 __ 5

(10) 1 b

22 5 4 1 b

18 5 b


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

15. (6, 1) and (18, 21)


m 5 y2 2 y1 _______ x2 2 x1

5 21 2 1 _______ 18 2 6

5 22 ___ 12

5 2 1 __ 6

Substitute m and b.

y 5 mx 1 b

y 5 2 1 __ 6

x 1 2


y 5 mx 1 b

1 5 2 1 __ 6

(6) 1 b

1 5 21 1 b

2 5 b

16. ( 2, 15 3 __ 4

) and ( 5, 39 3 __ 4

)


m 5 y2 2 y1 _______ x2 2 x1

5 39 3 __

4 2 15 3 __

4 __________

5 2 2

5 24 ___ 3

5 8

Substitute m and b.

y 5 mx 1 b

y 5 8x 2 1 __ 4


y 5 mx 1 b

15 3 __ 4 5 8(2) 1 b

15 3 __ 4 5 16 1 b

2 1 __ 4

5 b


© 2

011

Car

negi

e Le

arni

ng


17. (7, 1) and (21, 11)


m 5 y2 2 y1 _______ x2 2 x1

5 11 2 1 _______ 21 2 7

5 10 ___ 14

5 5 __ 7

Substitute m and b.

y 5 mx 1 b

y 5 5 __ 7

x 2 4


y 5 mx 1 b

1 5 5 __ 7

(7) 1 b

1 5 5 1 b

24 5 b

18. (2, 27) and (4, 210)


m 5 y2 2 y1 _______ x2 2 x1

5 210 2 (27)

___________ 4 2 2

5 23 ___ 2

Substitute m and b.

y 5 mx 1 b

y 5 2 3 __ 2

x 2 4


y 5 mx 1 b

27 5 2 3 __ 2 (2) 1 b

27 5 23 1 b

24 5 b


© 2

011

Car

negi

e Le

arni

ng


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)

9(x 2 2) 5 (y 2 21)

9x 2 18 5 y 2 21

9x 1 3 5 y

y 5 9x 1 3

20. slope 5 22 and (4, 7)

m(x 2 x1) 5 (y 2 y1)

22(x 2 4) 5 (y 2 7)

22x 1 8 5 y 2 7

22x 1 15 5 y

y 5 22x 1 15

21. slope 5 2 __ 5

and (10, 13)

m(x 2 x1) 5 (y 2 y1)

2 __ 5

(x 2 10) 5 (y 2 13)

2 __ 5

x 2 4 5 y 2 13

2 __ 5

x 1 9 5 y

y 5 2 __ 5

x 1 9

22. slope 5 2 4 __ 3

and (6, 216)

m(x 2 x1) 5 (y 2 y1)

2 4 __ 3

(x 2 6) 5 (y 2 (216))

2 4 __ 3

x 1 8 5 y 1 16

2 4 __ 3 x 2 8 5 y

y 5 2 4 __ 3

x 2 8


© 2

011

Car

negi

e Le

arni

ng


23. slope 5 1 __ 3

and (9, 212)

m(x 2 x1) 5 (y 2 y1)

1 __ 3 (x 2 9) 5 (y 2 (212))

1 __ 3 x 2 3 5 y 1 12

1 __ 3

x 2 15 5 y

y 5 1 __ 3 x 2 15

24. slope 5 2 5 ___ 12

and (24, 26)

m(x 2x1) 5 (y 2 y1)

2 5 ___ 12

(x 2 24) 5 (y 2 (26))

2 5 ___ 12

x 1 10 5 y 1 6

2 5 ___ 12

x 1 4 5 y

y 5 2 5 ___ 12

x 1 4

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

4x 16y 5 48

4(0) 1 6y 5 48

6y 5 48

y 5 8


4x 1 6y 5 48

4x 1 6(0) 5 48

4x 5 48

x 5 12

The x-intercept is (12, 0).

x

y

–16

–12

–8

–4

4

8

12

16

–4–8 4 8 12–12–16 16


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

26. 3x 115y 5 135

3x 1 15y 5 135

3(0) 1 15y 5 135

15y 5 135

y 5 9


3x 1 15y 5 135

3x 1 15(0) 5 135

3x 5 135

x 5 45


x

y

–8

–6

–4

–2

2

4

6

8

–10–20 10 20 30–30–40 40


© 2

011

Car

negi

e Le

arni

ng


27. 22x 1 8y 5 56

22x 1 8y 5 56

22(0) 1 8y 5 56

8y 5 56

y 5 7


22x 1 8y 5 56

22x 1 8(0) 5 56

22x 5 56

x 5 228


x

y

–8

–6

–4

–2

2

4

6

8

–8–16 8 16 24–24–32 32


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

28. 5x 16y 5 290

5x 1 6y 5 290

5(0) 1 6y 5 290

6y 5 290

y 5 215


5x 1 6y 5 290

5x 1 6(0) 5 290

5x 5 290

x 5 218


x

y

–16

–12

–8

–4

4

8

12

16

–4–8 4 8 12–12–16 16


© 2

011

Car

negi

e Le

arni

ng


29. 2x 1 8y 5 224

2x 1 8y 5 224

2(0) 1 8y 5 224

8y 5 224

y 5 23


2x 1 8y 5 224

2x 1 8(0) 5 224

2x 5 224

x 5 212


x

y

–16

–12

–8

–4

4

8

12

16

–4–8 4 8 12–12–16 16


© 2

011

Car

negi

e Le

arni

ng


Name ________________________________________________________ Date _________________________

30. 7x 2 3y 5 242

7x 2 3y 5 242

7(0) 2 3y 5 242

23y 5 242

y 5 14


7x 2 3y 5 242

7x 2 3(0) 5 242

7x 5 242

x 5 26


x

y

–16

–12

–8

–4

4

8

12

16

–4–8 4 8 12–12–16 16

name date · 540 600 length of track remaining in bobsled run 90 a b c point c is at (15, 150)....

Documents