Lesson 4.1 Skills Practice Name_________________________________________________________ Date__________________________
Willie Catchem Analyzing Problem Situations Using Multiple Representations
Problem Set Write an equation that represents the relationship shown in each table. Use the equation to answer the question. 1. Terell is biking around a track. The table show the time it takes Terell to complete several laps around the track. Write an equation that represents the relationship shown in the table. If Terell bikes around the track for 1 hour, how many laps will he complete? Time (minutes)
Distance (laps)
0
0
12
9
24
18
Let t 5 time (minutes) Let d 5 distance (laps) 9 laps speed 5 ___________ 12 minutes © 2011 Carnegie Learning
5 0.75 lap per minute d 5 0.75t
1 hour 5 60 minutes
d 5 0.75t
5 0.75(60)
5 45
If Terell bikes around the track for 1 hour, he will complete 45 laps. Chapter 4 Skills Practice • 467
Lesson 4.1 Skills Practice
page 2
2. Claudia is jogging around her neighborhood. The table shows the distance Claudia travels during her jog. Write an equation that represents the relationship shown in the table. How long will it take Claudia to jog 5 miles? Time (hours)
Distance (miles)
0
0
0.5
1.5
0.75
2.25
Let t 5 time (minutes) Let d 5 distance (miles) 1.5 miles speed 5 _________ 0.5 hour
5 3 miles per hour
d 5 3t
5 5 3t
__ 5 5 t
1.7 < t
It will take Claudia about 1.7 hours to jog 5 miles.
468 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
3
Lesson 4.1 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ 3. Shawna is driving to the beach for summer vacation. The table shows the distance Shawna travels during her trip. Write an equation that represents the relationship shown in the table. How long will it take Claudia to drive the entire 500 mile trip? Time (hours)
Distance (miles)
0
0
2
110
3
165
Let t 5 time (hours) Let d 5 distance (miles) 110 miles speed 5 _________ 2 hours
5 55 miles per hour d 5 55t
500 5 55t
© 2011 Carnegie Learning
t ____ 500 5 55
9.1 < t
It will take Shawna about 9.1 hours to travel 500 miles.
Chapter 4 Skills Practice • 469
Lesson 4.1 Skills Practice
page 4
4. Mattie and Lavonne agree to meet and make signs for the pep rally. Lavonne starts 30 minutes early and completes 2 signs before Mattie arrives. The table shows the total number of signs the girls have completed at different times after Mattie arrives. Write an equation that represents the relationship shown in the table. If the girls work for 3 hours after Mattie arrives, what is the total number of signs the girls will complete? Time (hours)
Number of Signs
20.5
0
0
2
0.5
4
Let t 5 time (hours) Let s 5 signs 4 2 2 rate 5 _______ 0.5 2 0
2 5 ___ 0.5
5 4 signs per hour s 5 2 1 4t
5 2 1 4(3)
5 14
If the girls work for 3 hours after Mattie arrives, they will have 14 total signs completed.
470 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
Lesson 4.1 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ 5. Juan leaves his home and rides his bike for 15 minutes to the park. After he gets to the park, he resets his odometer to 0 and continues to ride at the same speed. The table shows the distance Juan traveled at the park according to his odometer. Write an equation that represents the relationship shown in the table. If it has been 1.5 hours since Juan left home, what is the distance he rode his bike at the park? Time (hours)
Distance (miles)
0.25
0
0.5
1.25
0.75
2.5
Let t 5 time (hours) Let d 5 distance (miles) 1.25 2 0 speed 5 __________ 0.5 2 0.25
1.25 5 _____ 0.25
5 5 miles per hour
© 2011 Carnegie Learning
distance from Juan’s house to park 5 speed 3 time
5 (5)(0.25)
5 1.25
d 5 5t 2 1.25 5 5(1.5) 2 1.25 5 6.25 If it has been 1.5 hours since Juan left home, Juan rode his bike 6.25 miles at the park.
Chapter 4 Skills Practice • 471
Lesson 4.1 Skills Practice
page 6
6. Lee is a member of the drama club. He is setting up chairs in the gym for the school play. He arrives 5 minutes early and sets up 25 chairs before Jin arrives to help. The table shows the total number of chairs set up at different times after Jin arrives. Write an equation that represents the relationship shown in the table. If Lee and Jin continue at the same pace, how long will it take to set up 300 chairs? Time (minutes)
Number of Chairs
25
0
0
25
10
75
Let t 5 time (minutes) Let c 5 chairs 75 2 25 rate 5 ________ 10 2 0
50 5 ___ 10
5 5 chairs per minute
s 5 25 1 5t
275 5 5t 55 5 t If Lee and Jin continue to work together at the same pace, it will take 55 minutes to set up the 300 chairs.
472 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
300 5 25 1 5t
Lesson 4.1 Skills Practice
page 7
Name_________________________________________________________ Date__________________________ Complete each table using the given equation. Use the table to graph the equation. 7. Joelle is ice skating around an ice rink. The equation d 5 0.5t represents the relationship between the time Joelle skates in minutes, t, and the distance she skates in laps, d. Complete the table. Graph the points on the grid. Time (minutes)
Distance (laps)
0
0
10
5
20
10
30
15
50
y
40 30
Distance (laps)
20 10
-50
0 0 -40 -30 -20 -10 -10
x 10
20
30
40
50
© 2011 Carnegie Learning
-20 -30 -40 -50 Time (minutes)
Chapter 4 Skills Practice • 473
Lesson 4.1 Skills Practice
page 8
8. Leon is jogging around a park. The equation d 5 5t represents the relationship between the time Leon jogs in hours, t, and the distance he jogs in miles, d. Complete the table. Graph the points on the grid. Time (hours)
Distance (miles)
0
0
0.25
1.25
0.5
2.5
1
5
5
y
4 3
Distance (miles)
2 1 x
0 -2.5 -2.0 -1.5 -1.0 -0.5 -1
0
0.5
1.0
1.5
2.0
2.5
-2 -3
-5 Time (hours)
474 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
-4
Lesson 4.1 Skills Practice
page 9
Name_________________________________________________________ Date__________________________ 9. Perry is driving to college. The equation d 5 60t represents the relationship between the time Perry travels in hours, t, and the distance he travels in miles, d. Complete the table. Graph the points on the grid. Time (hours)
Distance (miles)
0
0
1
60
2
120
3
180
200
y
160 120
Distance (miles)
80 40 x
0 -5
-4
-3
-2
-1 -40
0
1
2
3
4
5
-80
© 2011 Carnegie Learning
-120 -160 -200 Time (hours)
Chapter 4 Skills Practice • 475
Lesson 4.1 Skills Practice
page 10
10. Elesha and Jada are making greeting cards to send to the children’s hospital. Jada makes 2 cards before Elesha arrives to help. The equation c 5 2 1 8t represents the relationship between the time Elesha and Jada make cards in hours, t, and the number of cards they make, c. Complete the table. Graph the points on the grid.
Time (hours)
Number of Cards
20.25
0
0
2
0.5
6
1
10
10
y
8 6
Number of Cards
4 2 x
0 -2.5 -2.0 -1.5 -1.0 -0.5 -2
0
0.5
1.0
1.5
2.0
2.5
-4 -6
-10 Time (hours)
476 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
-8
Lesson 4.1 Skills Practice
page 11
Name_________________________________________________________ Date__________________________ 11. Ling leaves her home and walks for 15 minutes to the park. After she gets to the park, she resets her pedometer to 0 and continues to walk at the same speed. The equation d 5 2t 2 0.5 represents the relationship between the time Li walks in hours, t, and the distance she walks in the park in miles, d. Complete the table. Graph the points on the grid. Time (hours)
Distance (miles)
0
20.5
0.25
0
0.5
0.5
0.75
1
2.5
y
2.0 1.5
Distance (miles)
1.0 0.5 x
0 -2.5 -2.0 -1.5 -1.0 -0.5 -0.5
0
0.5
1.0
1.5
2.0
2.5
© 2011 Carnegie Learning
-1.0 -1.5 -2.0 -2.5 Time (hours)
Chapter 4 Skills Practice • 477
© 2011 Carnegie Learning
478 • Chapter 4 Skills Practice
Lesson 4.2 Skills Practice Name_________________________________________________________ Date__________________________
Pony Express Interpreting the Standard Form of a Linear Equation
Problem Set Write an expression to represent each problem situation. Evaluate the expression to answer the question. 1. Aiko is walking 2 miles per hour. How far will she walk in 3 hours? Let x equal the number of hours Aiko walks. The expression 2x represents the distance Aiko walks in miles. 2x 5 2(3)
5 6 miles
Aiko will walk 6 miles in 3 hours.
2. Isabel is jogging 4 miles per hour. How long will it take her to jog 6 miles? Let x equal the number of hours Isabel jogs.
© 2011 Carnegie Learning
The expression 4x represents the distance Isabel jogs in miles. 4x 5 6 6 ___ __ 4x 5 4
4
x 5 1.5 It will take Isabel 1.5 hours to jog 6 miles.
Chapter 4 Skills Practice • 479
Lesson 4.2 Skills Practice
page 2
3. Chen is biking 6 miles per hour. How far will he travel in 2 hours? Let x equal the number of hours Chen bikes. The expression 6x represents the distance Chen bikes in miles. 6x 5 6(2)
5 12 miles
Chen will bike 12 miles in 2 hours.
4. Lamar is sketching people’s faces at the spring carnival. He can create 6 sketches per hour. How many sketches can he create in 4 hours? Let x equal the number of hours Lamar creates sketches. The expression 6x represents the number of sketches Lamar creates. 6x 5 6(4)
5 24 sketches
© 2011 Carnegie Learning
Lamar will create 24 sketches in 4 hours.
480 • Chapter 4 Skills Practice
Lesson 4.2 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ 5. Luis is shoveling snow from driveways in his neighborhood to save money for college. He can shovel 2 driveways per hour. How long will it take Luis to shovel 8 driveways? Let x equal the number of hours Luis shovels. The expression 2x represents number of driveways Luis shovels. 2x 5 8 8 ___ __ 2x 5 2
2
x54 It will take Luis 4 hours to shovel 8 driveways.
6. Darell is mowing lawns to save money for summer vacation. He can mow 1.5 lawns per hour. How many lawns can he mow in 6 hours? Let x equal the number of hours Darell mows lawns. The expression 1.5x represents the number of lawns Darell mows.
© 2011 Carnegie Learning
1.5x 5 1.5(6)
5 9 lawns
Darell will mow 9 lawns in 6 hours.
Chapter 4 Skills Practice • 481
Lesson 4.2 Skills Practice
page 4
Write an expression to represent each problem situation. 7. Eva and Sofia are participating in a relay for charity. Eva jogs 4 miles per hour. Sofia walks 2 miles per hour. Write an expression that represents the distance that Eva and Sofia travel during the relay. Let x equal the number of hours Eva jogs. The expression 4x represents the distance Eva jogs in miles. Let y equal the number of hours Sofia walks. The expression 2y represents the distance Sofia walks in miles. The expression 4x 1 2y represents the distance that Eva and Sofia travel during the relay.
8. Nina and Olivia are taking turns painting portraits at the spring carnival. Nina paints 3 portraits per hour. Olivia paints 2 portraits per hour. Write an expression that represents the number of portraits that Nina and Olivia paint during the carnival. Let x equal the number of hours Nina paints. The expression 3x represents the number of portraits Nina paints. Let y equal the number of hours Olivia paints. The expression 2y represents the number of portraits Olivia paints. The expression 3x 1 2y represents the number of portraits that Nina and Olivia paint during © 2011 Carnegie Learning
the carnival.
482 • Chapter 4 Skills Practice
Lesson 4.2 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ 9. Teresa and Juanita volunteer to take turns making signs for the football game. Teresa makes 8 signs per hour. Juanita makes 6 signs per hour. Write an expression that represents the number of signs that Teresa and Juanita make for the football game. Let x equal the number of hours Teresa makes signs. The expression 8x represents the number of sings Teresa makes. Let y equal the number of hours Juanita makes signs. The expression 6y represents the number of signs Juanita makes. The expression 8x 1 6y represents the number of signs that Teresa and Juanita make for the football game.
10. Geraldo and Raul are participating in a relay for charity. Geraldo rides his bike 6 miles per hour. Raul rides his bike 5 miles per hour. Write an expression that represents the distance that Geraldo and Raul travel during the relay. Let x equal the number of hours Geraldo rides his bike. The expression 6x represents the distance Geraldo rides in miles. Let y equal the number of hours Raul rides his bike. © 2011 Carnegie Learning
The expression 5y represents the distance Raul rides in miles. The expression 6x 1 5y represents the distance that Geraldo and Raul travel during the relay.
Chapter 4 Skills Practice • 483
Lesson 4.2 Skills Practice
page 6
11. Nelson and Ronna volunteer to take turns cleaning up the bike path at the park. Nelson cleans 1 mile per hour. Ronna cleans 2 miles per hour. Write an expression that represents the distance that Nelson and Ronna clean. Let x equal the number of hours Nelson cleans the bike path. The expression x represents the distance Nelson cleans in miles. Let y equal the number of hours Ronna cleans the bike path. The expression 2y represents the distance Ronna cleans in miles. The expression x 1 2y represents the distance that Nelson and Ronna clean.
12. Denisa and Tonya take turns making bead necklaces to sell at a school fundraiser. Denisa makes 2 necklaces per hour. Tonya makes 3 necklaces per hour. Write an expression that represents the number of necklaces that Denisa and Tonya make for the school fundraiser. Let x equal the number of hours Denisa makes necklaces. The expression 2x represents the number of necklaces Denisa makes. Let y equal the number of hours Tonya makes necklaces. The expression 3y represents the number of necklaces Tonya makes. The expression 2x 1 3y represents the number of necklaces that Denisa and Tonya make
© 2011 Carnegie Learning
for the school fundraiser.
484 • Chapter 4 Skills Practice
Lesson 4.2 Skills Practice
page 7
Name_________________________________________________________ Date__________________________ Write an equation to represent each problem situation. Solve the equation to answer the question. Check your answer. 13. The expression 3x 1 2y represents the distance that Shanise and Kiana travel during a 12 mile relay for charity. Let x represent the number of hours Shanise walks and let y represent the number of hours that Kiana walks. If Shanise walks for 2.5 hours, how many hours must Kiana walk?
Check: 3x 1 2y 5 12
3x 1 2y 5 12
3(2.5) 1 2y 5 12
3(2.5) 1 2(2.25) 0 12
7.5 1 2y 5 12
7.5 1 4.5 0 12
12 5 12
2y 5 4.5 y 5 2.25
© 2011 Carnegie Learning
Kiana must walk 2.25 hours.
Chapter 4 Skills Practice • 485
Lesson 4.2 Skills Practice
page 8
14. The expression 2x 1 4y represents the number of portraits that Nina and Olivia paint while taking turns at the spring carnival. They expect to paint 30 portraits at the carnival. Let x equal the number of hours Nina paints. Let y equal the number of hours Olivia paints. If Olivia paints for 4.5 hours, how long must Nina paint to complete the 30 portraits?
Check: 2x 1 4y 5 30
2x 1 4y 5 30
2x 1 4(4.5) 5 30
2(6) 1 4(4.5) 0 30
2x 1 18 5 30
12 118 0 30
2x 5 12
30 5 30
x56
Nina must paint for 6 hours.
15. The expression 5x 1 6y represents the number of signs that Joelle and Mattie make for the basketball game. They plan to take turns to make 25 signs. Let x equal the number of hours Joelle makes signs. Let y equal the number of hours Mattie makes signs. If Joelle makes signs for
Check:
5x 1 6y 5 25
5(2) 1 6y 5 25
5(2) 1 6(2.5) 0 25
10 1 6y 5 25
10 115 0 25
25 5 25
6y 5 15 y 5 2.5
Mattie must make signs for 2.5 hours.
486 • Chapter 4 Skills Practice
5x 1 6y 5 25
© 2011 Carnegie Learning
2 hours, how long must Mattie make signs?
Lesson 4.2 Skills Practice
page 9
Name_________________________________________________________ Date__________________________ 16. The expression 6x 1 5.5y represents the distance that Carlos and Leon travel during a 20 mile relay for charity. Let x equal the number of hours Carlos rides his bike. Let y equal the number of hours Leon rides his bike. If Leon rides his bike for 2 hours, how long must Carlos ride his bike to complete the relay?
Check:
6x 1 5.5y 5 20
6x 1 5.5(2) 5 20
6(1.5) 1 5.5(2) 0 20
9 1 11 0 20
20 5 20
6x 1 11 5 20 6x 5 9
6x 1 5.5y 5 20
x 5 1.5
© 2011 Carnegie Learning
Carlos must ride his bike for 1.5 hours.
Chapter 4 Skills Practice • 487
Lesson 4.2 Skills Practice
page 10
17. The expression 1.5x 1 2y represents the distance that Perry and Ty cover while volunteering to clean up the bike path at the park. The bike path is 8 miles long. Let x equal the number of hours Perry cleans the bike path. Let y equal the number of hours Ty cleans the bike path. If Perry cleans for 3 hours, how long must Ty clean?
Check:
1.5x 1 2y 5 8
1.5(3) 1 2y 5 8
1.5(3) 1 2(1.75) 0 8
4.5 1 2y 5 8
4.5 1 3.5 0 8
858
2y 5 3.5
1.5x 1 2y 5 8
y 5 1.75
Ty must clean the bike path for 1.75 hours.
18. The expression 4x 1 3.5y represents the number of necklaces that Denisa and Tonya make for the school fundraiser. They plan to make 30 necklaces. Let x equal the number of hours Denisa makes necklaces. Let y equal the number of hours Tonya makes necklaces. If Tonya makes necklaces for
Check:
4x 1 3.5y 5 30
4x 1 3.5y 5 30
4x 1 3.5(4) 5 30
4(4) 1 3.5(4) 0 30
4x 1 14 5 30
16 1 14 0 30
4x 5 16
30 5 30
x54
Denisa must make necklaces for 4 hours.
488 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
4 hours, how long must Denisa make necklaces?
Lesson 4.3 Skills Practice Name_________________________________________________________ Date__________________________
Slopes, Forms, Graphs, and Intercepts Connecting the Standard Form with the Slope-Intercept Form of Linear Functions
Problem Set Complete each table using the equation in standard form to determine the x-intercept and y-intercept. Calculate the slope. Graph the equation. 1. 2x 1 4y 5 12
x
y
Work 2x 1 4y 5 12 2(0) 1 4y 5 12
0
3 4y 5 12 y 5 3 2x 1 4y 5 12 2x 1 4(0) 5 12
6
0
© 2011 Carnegie Learning
2x 5 12 x 5 6
Chapter 4 Skills Practice • 489
Lesson 4.3 Skills Practice
page 2
y-intercept: (0, 3)
10
y
8
x-intercept: (6, 0)
6
slope 5 ______ 3 2 0 026
4 2
3 5 ___ 26
x
0 -10
5 2__ 1 2
-8
-6
-4
0
-2
2
4
6
8
10
-2 -4 -6 -8 -10
2. 3x 2 5y 5 30
x
y
Work 3x 2 5y 5 30 3(0) 2 5y 5 30
0
26
y 5 26 3x 2 5y 5 30 3x 2 5(0) 5 30 10
0 3x 5 30 x 5 10
490 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
25y 5 30
Lesson 4.3 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ y-intercept: (0, 26)
10
y
8
x-intercept: (10, 0)
6
0 2 (26) slope 5 ________
4
10 2 0
2
6 5 ___ 10
x
0 -10
3 5 __ 5
-8
-6
-4
0
-2
2
4
6
8
10
-2 -4 -6 -8 -10
3. 4x 1 2y 5 14
x
y
Work 4x 1 2y 5 14 4(0) 1 2y 5 14
© 2011 Carnegie Learning
0
7 2y 5 14 y 5 7 4x 1 2y 5 14 4x 1 2(0) 5 14
3.5
0 4x 5 14 x 5 3.5
Chapter 4 Skills Practice • 491
Lesson 4.3 Skills Practice
page 4
y-intercept: (0, 7)
10 8
x-intercept: (3.5, 0)
6
0 2 7 slope 5 _______
4
3.5 2 0
27 5 ___ 3.5
5 22
y
2 x
0 -10
-8
-6
-4
0
-2
2
4
6
8
10
-2 -4 -6 -8 -10
4. 3x 2 4y 5 224
x
y
Work 3x 2 4y 5 224 3(0) 2 4y 5 224
0
6
y 5 6 3x 2 4y 5 224 3x 2 4(0) 5 224 28
0 3x 5 224 x 5 28
492 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
24y 5 224
Lesson 4.3 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ y-intercept: (0, 6)
10
y
8
x-intercept: (28, 0)
6
0 2 6 slope 5 _______
4
28 2 0
2
26 5 ___ 28
x
0 -10
3 5 __ 4
-8
-6
-4
0
-2
2
4
6
8
10
-2 -4 -6 -8 -10
5. 22x 1 3y 5 27.5
x
y
Work 22x 1 3y 5 27.5 22(0) 1 3y 5 27.5
© 2011 Carnegie Learning
0
22.5 3y 5 27.5 y 5 22.5 22x 1 3y 5 27.5 22x 1 3(0) 5 27.5
3.75
0 22x 5 27.5 x 5 3.75
Chapter 4 Skills Practice • 493
Lesson 4.3 Skills Practice
page 6
y-intercept: (0, 22.5)
10
y
8
x-intercept: (3.75, 0)
6
0 2 (22.5) slope 5 __________ 3.75 2 0
2.5 5 _____ 3.75
2 5 __ 3
4 2 x
0 -10
-8
-6
-4
0
-2
2
4
6
8
10
-2 -4 -6 -8 -10
6. 2x 2 0.5y 5 4
x
y
Work 2x 2 0.5y 5 4 2(0) 2 0.5y 5 4
0
28
y 5 28 2x 2 0.5y 5 4 2x 2 0.5(0) 5 4 2
0 2x 5 4 x 5 2
494 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
20.5y 5 4
Lesson 4.3 Skills Practice
page 7
Name_________________________________________________________ Date__________________________ y-intercept: (0, 28)
10 8
x-intercept: (2, 0)
6
0 2 (28) slope 5 ________
4
220
y
2
8 5 __ 2
x
0 -10
54
-8
-6
-4
0
-2
2
4
6
8
10
-2 -4 -6 -8 -10
Convert each equation in standard form to slope-intercept form. 8. 2x 1 8y 5 216
7. 5x 1 6y 5 60
© 2011 Carnegie Learning
5x 1 6y 5 60
2x 1 8y 5 216
5x 2 5x 1 6y 5 25x 1 60
2x 2 2x 1 8y 5 22x 2 16
6y 5 25x 1 60
8y 5 22x 2 16
6y _________ 60 ___ 25x 1 5
8y _________ 16 ___ 22x 2 5
6
6
5 y 5 2 __ x 1 10 6
8
8
1 y 5 2 __ x 2 2 4
Chapter 4 Skills Practice • 495
Lesson 4.3 Skills Practice
9. 210x 1 15y 5 45
page 8
10. 8x 1 2y 5 14
210x 1 15y 5 45
8x 1 2y 514
210x 1 10x 1 15y 5 10x 1 45
8x 2 8x 1 2y 5 28x 1 14
15y 5 10x 1 45
2y 5 28x 1 14
15y _________ 45 ____ 10x 1 5
2y _________ ___ 28x 114 5
15
15
2 x 1 3 y 5 __ 3
11. 18x 2 3y 5 6
2
2
y 5 24x 1 7
12. 3x 1 2y 5 10
18x 2 3y 5 6
3x 1 2y 5 10
18x 2 18x 2 3y 5 218x 1 6
3x 2 3x 1 2y 5 23x 1 10
23y 5 218x 1 6
2y 5 23x 1 10
23y _________ 6 ____ 218x 1 5
2y _________ 10 23x 1 ___ 5 2 2
23
y 5 6x 2 2
3 y 5 2 __ x 1 5 2 © 2011 Carnegie Learning
23
496 • Chapter 4 Skills Practice
Lesson 4.3 Skills Practice
page 9
Name_________________________________________________________ Date__________________________ Convert each equation in slope-intercept form to standard form. 13. y 5 __ 1 x 1 9 4
14. y 5 28x 1 3
1 x 1 9 y 5 __ 4
(
)
1 x 1 9 4y 5 4__ 4
4y 5 x 1 36
y 5 28x 1 3
8x 1 y 5 28x 1 8x 1 3 8x 1 y 5 3
2x 1 4y 5 x 2 x 1 36 2x 1 4y 5 36
15. y 5 3x 2 2
y 5 3x 2 2
23x 1 y 5 3x 2 3x 2 2
© 2011 Carnegie Learning
23x 1 y 5 22
2 16. y 5 2__ x 1 1 3
2 x 1 1 y 5 2 __ 3
(
)
2 x 1 1 3y 5 3 2__ 3
3y 5 22x 1 3
2x 1 3y 5 22x 1 2x 1 3 2x 1 3y 5 3
Chapter 4 Skills Practice • 497
Lesson 4.3 Skills Practice 5 17. y 5 2__ x 2 6 2
18. y 5 29x 2 2
5 y 5 2 __ x 2 6 2
5 x 26 2y 5 2 2__ 2
2y 5 25x 212
(
page 10
)
y 5 29x 2 2
9x 1 y 5 29x 1 9x 2 2 9x 1 y 5 22
5x 1 2y 5 25x 1 5x 212
© 2011 Carnegie Learning
5x 1 2y 5 212
498 • Chapter 4 Skills Practice
Lesson 4.4 Skills Practice Name_________________________________________________________ Date__________________________
The Journey Starts with a Single Step—But There Are Many Steps After That! Intervals of Increase, Decrease, and No Change
Vocabulary Choose the term or phrase from the box that correctly completes each sentence. constant function
increasing function
decreasing function
interval of increase
interval of decrease
constant interval
1. When the value of a dependent variable decreases as the independent variable increases, the function is said to be a(n) decreasing function. 2. When both the dependent and independent values of a function increase, the function is said to be a(n) increasing function . 3. When a function is decreasing for some values of the independent variable, it is said to have a(n) interval of decrease. 4. When the dependent variable does not change as the independent value of a function increases, the function is said to be a(n) constant function .
© 2011 Carnegie Learning
5. When a function is constant for some values of the independent variable, it is said to have a(n) constant interval . 6. When a function is increasing for some values of the independent variable, it is said to have a(n) interval of increase .
Chapter 4 Skills Practice • 499
Lesson 4.4 Skills Practice
page 2
Problem Set Identify each function displayed in each graph as an increasing function, a decreasing function, or a constant function. 2. y
1. y
x
x
decreasing function
constant function
3. y
© 2011 Carnegie Learning
4. y
x
increasing function
500 • Chapter 4 Skills Practice
x
increasing function
Lesson 4.4 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ 5. y
6. y
x
constant function
x
decreasing function
Identify each interval of increase, the interval of decrease, and the constant interval as appropriate for the function shown in each graph. 7.
10
y
Constant interval: The function is constant from 0 to 3.
9
Interval of increase: The function is
© 2011 Carnegie Learning
8
increasing from 3 to 5.5.
7
Interval of decrease: The function is
6
decreasing from 5.5 to 8.5.
5 4 3 2 1 0 0
1
2
3
4
5
6
7
8
9
x 10
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Lesson 4.4 Skills Practice
8.
100
page 4
Interval of increase: The function is
y
increasing from 0 to 20.
90
Constant interval: The function is
80
constant from 20 to 65.
70
Interval of decrease: The function is
60
decreasing from 65 to 95.
50 40 30 20 10 0 0
10
20
30
40
9.
50
10
60
70
80
90
x 100
y
decreasing until 4.
8
Interval of decrease: The function is Interval of increase: The function is
6
increasing after 4.
4 y = |x - 4|
2
x
0 -10
-8
-6
-4
0
-2
2
4
6
8
10
-4 -6 -8 -10
502 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
-2
Lesson 4.4 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ 10.
10
y
Interval of increase: The function is increasing until 22.
8
Interval of decrease: The function is
6
decreasing after 22.
4 2 x
0 -10
-8
-6
-4
0
-2
2
4
6
8
10
-2 y = -|x + 2|
-4 -6 -8 -10
11.
20
Interval of decrease: The function is
y
decreasing from 0 to 5.
© 2011 Carnegie Learning
18
Constant interval: The function is constant
16
from 5 to 11.
14
Interval of increase: The function is
12
increasing from 11 to 15.
10 8 6 4 2 0 0
2
4
6
8
10
12
14
16
18
x 20
Chapter 4 Skills Practice • 503
Lesson 4.4 Skills Practice
12.
40
page 6
Interval of increase: The function is
y
increasing from 0 to 12.
36
Interval of decrease: The function is
32
decreasing from 12 to 22.
28
Constant interval: The function is constant
24
from 22 to 38.
20 16 12 8 4 x 40
0 0
4
8
12
16
20
24
28
32
36
Define variables and write an equation (or equations) to represent the problem situation. Graph the equation (or equations). 13. Noah walks from home to school at a rate of 5 feet per second. It takes Noah 15 minutes to walk to school. Write an equation that represents Noah’s distance from home. Graph the equation. Let t 5 time in seconds
5000
y
4500
Let d 5 distance from home in feet
5 900 seconds
d 5 5t d 5 5(900)
3500 3000 © 2011 Carnegie Learning
15 minutes 5 15(60)
Distance From Home (in feet)
4000
2500 2000 1500 1000
d 5 4500
500 0 0
504 • Chapter 4 Skills Practice
x 100 200 300 400 500 600 700 800 900 1000 Time (in seconds)
Lesson 4.4 Skills Practice
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Name_________________________________________________________ Date__________________________ 14. Mario is at school, 3500 feet from his home. He stays there for 2 hours. Write an equation that represents Mario’s distance from home. Graph the equation. Let t 5 time in hours
5000
y
4500
Let d 5 distance from home in feet
d 5 3500
Distance From Home (in feet)
4000 3500 3000 2500 2000 1500 1000 500 x
0 1
2 3 Time (in hours)
4
5
© 2011 Carnegie Learning
0
Chapter 4 Skills Practice • 505
Lesson 4.4 Skills Practice
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15. Hector is at school, 7200 feet from home. He leaves school and rides his bike home at a rate of 12 feet per second. Write an equation that represents Hector’s distance from home. Graph the equation. Let t 5 time in seconds 8000
Let d 5 distance from home in feet
from home: (0, 7200) 7200 ft 12 ft 5 __________ _________ 1 second
? seconds
12x 5 7200 x 5 600 seconds
Hector arrives home after 600 seconds: (600, 0) 0 2 7200 m 5 __________ 600 2 0
7200 6400 Distance From Home (in feet)
At 0 seconds Hector is 7200 feet
y
5600 4800 4000 3200 2400 1600 800 0 0
x 100 200 300 400 500 600 700 800 900 1000 Time (in seconds)
27200 m 5 _______ 600 m 5 212
d 5 212t 1 7200
506 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
y 5 mx 1 b
Lesson 4.4 Skills Practice
page 9
Name_________________________________________________________ Date__________________________ 16. Rita walks from home to the store at a rate of 6 feet per second. It takes her 10 minutes to walk to the store. After arriving at the store, she stays for 30 minutes. Write an equation that represents Rita’s distance from home. Graph the equation. Let t 5 time in seconds 4000
Let d 5 distance from home in feet
5 600 seconds
d 5 6t d 5 6(600)
3600 3200 Distance From Home (in feet)
10 minutes 5 10(60)
y
2800 2400 2000 1600 1200 800
d 53600
400
For the interval of increase from t 5 0
0
to t 5 600, Rita’s distance from home
0
x 400 800 1200 1600 2000 2400 2800 3200 3600 4000 Time (in seconds)
can be represented by d 5 6t. 30 minutes 5 30(60)
© 2011 Carnegie Learning
5 1800 seconds
600 seconds 1 1800 seconds 5 2400 seconds For the constant interval from t 5 600 seconds to t 5 2400 seconds, Rita’s distance from home can be represented by d 5 3600.
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Lesson 4.4 Skills Practice
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17. Elena is at Lea’s house, which is 3600 feet from Elena’s home. Elena leaves Lea’s house and walks in the direction toward her home at a rate of 5 feet per second. After 8 minutes, she stops at Cristina’s house. She stays at Cristina’s house for 30 minutes. Write an equation that represents Elena’s distance from home. Graph the equation. Let t 5 time in seconds 4000
Let d 5 distance from home in feet
home: (0, 3600) 8 minutes 5 8(60)
5 480 seconds
5 feet 2400 feet 5 480 seconds 3 _________
3600 3200 Distance From Home (in feet)
At 0 seconds, Elena is 3600 feet from
y
2800 2400 2000 1600 1200
1 second
3600 feet 2 2400 feet 5 1200 feet
800 400 0
Elena arrives at Cristina’s house at (480, 1200)
0
x 400 800 1200 1600 2000 2400 2800 3200 3600 4000 Time (in seconds)
y 5 mx 1 b d 5 25t 1 3600 For the interval of decrease from t 5 0 to t 5 480, Elena’s distance from home can
30 minutes 5 30(60)
5 1800 seconds
480 seconds 1 1800 seconds 5 2280 seconds For the constant interval from t 5 480 seconds to t 5 2280 seconds, Rita’s distance from home can be represented by d 5 1200.
508 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
be represented by d 5 25t 1 3600.
Lesson 4.4 Skills Practice
page 11
Name_________________________________________________________ Date__________________________ 18. Nina is at her friend Maria’s house 9000 feet from home. She stays at Maria’s house for 1 hour. Nina then rides her bike home at a rate of 10 feet per second. Write an equation that represents Nina’s distance from home. Graph the equation. Let t 5 time in seconds 9000
Let d 5 distance from home in feet
from home: (0, 9000) 1 hour 5 1(60)(60) 5 3600 seconds
At 3600 seconds, Nina is still 9000 feet
8100 7200 Distance From Home (in feet)
At 0 seconds, Nina is 9000 feet
y
6300 5400 4500 3600 2700 1800
from home: (3600, 9000)
900
For the constant interval from t 5 0 seconds to t 5 3600 seconds, Nina’s distance from
0 0
x 450 900 1350 1800 2250 2700 3150 3600 4050 4500 Time (in seconds)
home can be represented by d 5 9000. 9000 ft _________ __________ 10 ft 5 1 second
© 2011 Carnegie Learning
? seconds
10x 5 9000
x 5 900 seconds
Nina arrives home after 900 seconds: (4500, 0) (y 2 y1) 5 m(x 2 x1) (y 2 0) 5 210(x 2 4500)
y 5 210x 1 45,000
For the interval of decrease from t 5 3600 to t 5 4500, Nina’s distance from home can be represented by d 5 210t 1 45,000.
Chapter 4 Skills Practice • 509
© 2011 Carnegie Learning
510 • Chapter 4 Skills Practice
Lesson 4.5 Skills Practice Name_________________________________________________________ Date__________________________
Piecewise Functions Developing the Graph of a Piecewise Function
Vocabulary Define the term in your own words. 1. piecewise function A piecewise function is a function that can be represented by more than one function, each of which corresponds to a part of the domain.
Problem Set Write a piecewise function from each table or context. 1.
Minutes
0
1
10
20
40
41
50
Gallons of Water in Tub
0
2
20
40
40
36
0
2x, 0 # x # 20 f(x) 5 40, 20 , x # 40
© 2011 Carnegie Learning
200 2 4x, 40 , x # 50
Chapter 4 Skills Practice • 511
Lesson 4.5 Skills Practice
page 2
2. At 10:00, Sheila biked to Jill’s house. She stayed at Jill’s for an hour. Then, she biked back home. Sheila can bike 10 miles per hour uphill and 20 miles per hour downhill. (Hint: Use minutes as the x-value.)
Jill’s house 10 Miles
Sheila’s house
1 x, 0 # x # 60 __ 6 f(x) 5 10, 60 , x # 120 1 2 __ x 1 50, 120 , x # 150 3
3. Kyle saved $4 per day for 30 days. Then, he spent $5 a day for 8 days. After that he started saving again at a rate of $3 per day for 22 days.
4x, 0 # x # 30 f(x) 5 25x 1 270, 30 , x # 38
4.
Hours
0
2
5
6
7
8
10
Miles Driven
0
140
350
395
460
525
655
70x, 0 # x # 5 f(x) 5 45x 1 125, 5 , x # 6 65x 1 5, 6 , x # 10
512 • Chapter 4 Skills Practice
© 2011 Carnegie Learning
3x 2 34, 38 , x # 60
Lesson 4.5 Skills Practice
page 3
Name_________________________________________________________ Date__________________________ 5. Miles received a $50 gift card to an online music store from his aunt. He decided to spend a certain amount each day to extend his enjoyment of the gift card. For the first three days, he set his limit for $5 per day. Then, he was worried it wouldn’t last long enough at that rate, so he changed his limit to $3 per day until he had $2 left. Then, he used the last $2 on the final day.
25x 1 50, 0 # x # 3 f(x) 5 23x 1 44, 3 , x # 14 22x 1 30, 14 , x # 15
6.
Seconds
5
10
15
16
20
21
30
31
50
Jump Rope Jumps
15
30
45
49
65
70
115
118
175
3x, 0 # x # 15 4x 2 15, 15 , x # 20 f(x) 5 5x 2 35, 20 , x # 30 3x 1 25, 30 , x # 50
© 2011 Carnegie Learning
7. Kara is a triathlete. It took her 15 minutes to swim, 80 minutes to bike, and 40 minutes to run one race.
1.5 km
40 km
10 km
0.1x, 0 # x # 15 f(x) 5 0.5x 2 6, 15 , x # 95 0.25x 1 17.75, 95 , x # 135
Chapter 4 Skills Practice • 513
Lesson 4.5 Skills Practice
8.
page 4
Days
0
5
10
17
18
20
21
Calling Card Minutes
100
70
40
40
30
10
0
26x 1 100, 0 # x # 10 f(x) 5 40, 10 , x # 17 210x 1 210, 17 , x # 21
Graph each piecewise function from the table or context. 9.
Minutes
0
1
10
20
40
41
50
Gallons of Water in Tub
0
2
20
40
40
36
0
Variable Quantity
Upper Bound
Lower Bound
Interval
Minutes (x)
50
0
5
Galons of Water in Tub f(x)
50
0
5
y 50 45
© 2011 Carnegie Learning
Gallons of Water in Tub
40 35 30 25 20 15 10 5 0
5
10
15
20
25
30
35
Minutes
514 • Chapter 4 Skills Practice
40
45
50
x
Lesson 4.5 Skills Practice
page 5
Name_________________________________________________________ Date__________________________ 10. At 10:00, Sheila biked to Jill’s house. She stayed at Jill’s for an hour. Then, she biked back home. Sheila can bike 10 miles per hour uphill and 20 miles per hour downhill. (Hint: Use minutes as the x-value.)
Jill’s house 10 Miles
Sheila’s house
Variable Quantity
Upper Bound
Lower Bound
Interval
Minutes (x)
150
0
15
Distance from Home f(x)
10
0
1
y 10 9
Distance from Home
© 2011 Carnegie Learning
8 7 6 5 4 3 2
1 0
15
30
45
60
75
90 105 120 135 150
x
Minutes
Chapter 4 Skills Practice • 515
Lesson 4.5 Skills Practice
page 6
11. Kyle saved $4 a day for 30 days. Then, he spent $5 a day for 8 days. After that, he started saving again at a rate of $3 a day for 22 days.
Variable Quantity
Upper Bound
Lower Bound
Interval
Days (x)
60
0
10
Money f(x)
150
0
25
y 150
Money
125 100 75 50 25
0
20
30
40
50
60
x
Days
© 2011 Carnegie Learning
10
516 • Chapter 4 Skills Practice
Lesson 4.5 Skills Practice
page 7
Name_________________________________________________________ Date__________________________ 12.
Hours
0
2
5
6
7
8
10
Miles Driven
0
140
350
395
460
525
655
Variable Quantity
Upper Bound
Lower Bound
Interval
Hours (x)
10
0
1
Miles f(x)
700
0
100
y 700 600
Miles
500 400 300 200 100
© 2011 Carnegie Learning
0
1
2
3
4
5
6
7
8
9
10
x
Hours
Chapter 4 Skills Practice • 517
Lesson 4.5 Skills Practice
page 8
13. Miles received a $50 gift card to an online music store from his aunt. He decided to spend a certain amount each day to extend his enjoyment of the gift card. For the first three days, he set his limit for $5 a day. Then, he was worried it wouldn’t last long enough at that rate, so he changed his limit to $3 a day until he had $2 left. Then he used the last $2 on the last day.
Variable Quantity
Upper Bound
Lower Bound
Interval
Days (x)
20
0
3
Money f(x)
50
0
5
y 50 45 40
Money
35 30 25 20 15 10 5 0
6
9
12
15
18
x
Days
© 2011 Carnegie Learning
3
518 • Chapter 4 Skills Practice
Lesson 4.5 Skills Practice
page 9
Name_________________________________________________________ Date__________________________ 14.
Seconds
5
10
15
16
20
21
30
31
50
Jump Rope Jumps
15
30
45
49
65
70
115
118
175
Variable Quantity
Upper Bound
Lower Bound
Interval
Seconds (x)
50
0
5
Jumps f(x)
250
0
25
y 250 225 200
Jumps
175 150 125 100 75 50 25 0
10
15
20
25
30
35
40
45
50
x
Seconds
© 2011 Carnegie Learning
5
Chapter 4 Skills Practice • 519
Lesson 4.5 Skills Practice
page 10
15. Kara is a triathlete. It took her 15 minutes to swim, 80 minutes to bike, and 40 minutes to run one race.
1.5 km
40 km
10 km
Variable Quantity
Upper Bound
Lower Bound
Interval
Minutes (x)
200
0
20
Kilometers f(x)
55
0
5
y 55 50 45
Kilometers
40 35 30 25 20 15
5 0
20
40
60
80 100 120 140 160 180 200 Minutes
520 • Chapter 4 Skills Practice
x
© 2011 Carnegie Learning
10
Lesson 4.5 Skills Practice
page 11
Name_________________________________________________________ Date__________________________ 16.
Days
0
5
10
17
18
20
21
Calling Card Minutes
100
70
40
40
30
10
0
Variable Quantity
Upper Bound
Lower Bound
Interval
Days (x)
21
0
5
Minutes f(x)
100
0
10
y 100 90
Calling Card Minutes
80 70 60 50 40 30 20 10
© 2011 Carnegie Learning
0
5
10
15
20
x
Days
Chapter 4 Skills Practice • 521
© 2011 Carnegie Learning
522 • Chapter 4 Skills Practice
