FUNCTIONS - RUSD Mathrusdmath.weebly.com/uploads/1/1/1/5/11156667/g8_u7... · 2018-10-14 · Day Tickets and Full Festival Tickets Number of Tickets (in 1,000s) Income (i n $1,0 00

ANSWERS FOR EXERCISES

MATH GRADE 8 UNIT 7

FUNCTIONS

Copyright © 2015 Pearson Education, Inc. 34

Grade 8 Unit 7: Functions

ANSWERS

ANSWERS

1. A Increasing and nonlinear

2. D –4 to 0

3. a. x (input) –4 –3 –2 –1 0 1 2 3 4

y (output) –

14

–13

–12

–1 Not defined 1

12

13

14

b. Answers will vary. Possible answer: There are no definite maxima or minima points. From negative infinity to x = 0 the left side of the graph is decreasing. The graph approaches x = 0 but never reaches a value at x = 0. The graph decreases slowly at first but then decreases rapidly as it approaches x = –1. The inverse is true for the right side of the graph. From x = 0 to positive infinity the right side of the graph is decreasing. The graph decreases rapidly at first but then decreases slowly as it approaches x = 1.

4. a. Answers will vary. Possible answer: Federer was the best tennis player at the time. The graph shows that at age 24 Federer surpassed Nadal and Djokovic in the number of Grand Slam titles won.

b. Starting at age 25 Nadal’s graph slows down to a constant rate (not winning any more Grand Slam titles) while Federer’s graph continues to rise at a steady rate.

c. When players win Grand Slam titles their graph increases. When players do not win more Grand Slam titles their graph remains constant. The graph would never decrease because titles would not be taken away.

Challenge Problem

5. Answers will vary. Ask a classmate to check your work.

LESSON 2: WHAT A GRAPH CAN TELL YOU



ANSWERS

ANSWERS

1. B (1, 3), (2, 6), (3, 9)

2. D Line D

3. a. The truck passes the bike after 9 sec.

b. Using the points (2, 40) and (9, 140) you can determine the speed of the bike

by finding the slope of the line. The slope is 1007 , or about 14.286 ft/sec.

= = …140 – 409 – 2

1007

14.285714 = 14.285714

c. The speed of the new bike traveling at 10 mph needs to be converted into feet per second.

1 hr

3,600 sec5,280 ft

1 mi10 mi1 hr

• = 14.666… or about 14.667 ft/sec• = 14.6 or about 14.667 ft/sec

This speed is slightly faster than that of the previous bike, which has a speed of 14.286 ft/sec.

d. The third bike starts farther ahead of the other two, starting at 20 ft. But it travels significantly slower, at only 10 ft/sec.

e. At around 5 sec the slope of the truck’s curve is about the same as the slope of the bike’s line. This fact indicates that they are at about the same speed.

LESSON 3: LINEAR VERSUS NONLINEAR GRAPHS



ANSWERSLESSON 3: LINEAR VERSUS NONLINEAR GRAPHS

4. Since the lines are parallel you know they have the same slope. The slope of the

second line (described by the two points) can be calculated as –32

.

= =6 – 00 – 4

–64

–32

So, the slope of the first line must be the same. Thus, you get the equation

y = –32

x + 4 for the first line and a = –32

for the second line.

1

2

3

4

5

6

7

8

–3

–2

–154321–1–2–3–4–5 x

y

Challenge Problem

5. a. High tide is usually around 4 m and low tide is roughly 1 m.

b. There is a little more than 12 hr between each peak on the graph—approximately 12.5 hr.

c.

4 6 8 10 12a.m.

2 4 6 8 10 12p.m.

2 4 6 8 10 12a.m.

2 4 6 8

5

4

3

2

1

0

Time (hr)

Hei

ght

(m)

2



ANSWERS

ANSWERS

1. C (1, 3), (1, 4), (1, 5)

2. D 22

3. a. 30°C is the same as 86°F.

f = 95

(30) + 32

f = 54 + 32 f = 86

b. Yes, f = 95

c + 32 is a function. The equation represents a function because it is

a line and for any given input (c-value) there is exactly one associated output

(f-value).

c.

10

20

30

40

50

50

60

70

80

90

100

4020 3010–20 x

y

Temperature Conversion from Degrees Celsius to Degrees Fahrenheit

Tem

pera

ture

in F

ahr

enhe

it (F

°)

Temperature in Celsius (C°)

= +( )95

• 32f x x

(continues)

LESSON 4: WHAT IS A FUNCTION?



ANSWERSLESSON 4: WHAT IS A FUNCTION?

(continued)

d. The constant rate of change is 95

; this rate means that for every increase of 9°F

there is an increase of 5°C.

e. Yes, the dots can be connected because the graph is continuous. You could measure a temperature of 50.5º and that would still make sense.

4. The formula for converting from Fahrenheit to Celsius is c = (f – 32).

To convert a temperature of 100°F to degrees Celsius:

c = 59

⎛⎝⎜

⎛⎝⎜(100 – 32)

c = 59

⎛⎝⎜

⎛⎝⎜(68)

c = 37.7

Challenge Problem

5. Answers will vary. Share your findings with a classmate.

The equations involved are shown in the table.

From Kelvin (K) To Kelvin (K)

Celsius (C) °C = K − 273.15 K = °C + 273.15

Fahrenheit (F) °F = K • 95

− 459.67 K = (°F + 459.67) • 59

59



ANSWERS

ANSWERS

1. B 4

2. C y = 3x + 18

3. a. At t = 0 Talisha starts 30 ft closer to the finish line than Erin.

70 – 40 = 30

b. Answers will vary. Possible answer: Erin starts 70 ft from the finish line. She runs at 5 ft/sec the entire race. She finishes the race after 14 sec. Talisha starts only 40 ft away from the finish line. She runs at 2 ft/sec for the entire race. Talisha finishes the race after 20 sec. Even though Erin starts significantly behind Talisha, Erin passes Talisha after 10 sec and wins the race!

c. Answers will vary. Possible answer: The boys’ story is quite similar to the girls’ story. Marshall starts 40 ft from the finish, runs at 4 ft/sec for the entire race, and completes the race after 10 sec. Jacob starts the race 22 ft from the finish, runs at 1 ft/sec, and finishes the race after 22 sec. Marshall passes Jacob after 6 sec and wins the race.

d. Here are all four runners’ speeds: Erin: 5 ft/sec

Talisha: 2 ft/sec

Marshall: 4 ft/sec

Jacob: 1 ft/sec

Erin is the fastest, followed by Marshall, then Talisha, and finally Jacob.

e. Erin: d = –5t + 70

Talisha: d = –2t + 40

Marshall: d = –4t + 40

Jacob: d = –1t + 22

(continues)

LESSON 6: WHAT IS A LINEAR FUNCTION?



ANSWERSLESSON 6: WHAT IS A LINEAR FUNCTION?

(continued)

f. The girls’ graphs intersect at (10, 20).

The boys’ graphs intersect at (6, 16).

Girls: Boys:

Challenge Problem

4. a.

1

0

2

3

4

5

6

7

8

9

10

12

4 6 820 x

y Taxi Fares

Distance (mi)

Cos

t ($

)

b. Answers will vary. Possible answer: The first section indicates that for any distance between 0 mi and 2 mi, you will pay $4.00. After that, the fare goes up pretty steeply, at a rate of $0.35/0.2 mi.

The slope is 74

. For example, if you go 6 mi your total fare is now $11.00.

If you bring 3 friends, the entire graph shifts up vertically by $3.00, since the charge is $1.00 per each extra rider, no matter the distance.

( )

( )

( )= += += +=

= += +=

–5 70

– – –2 40

0 –3 30

10

–5 70

–5 10 70

20

d t

d t

t

t

d t

d

d

( )

( )

( )= += += +=

= += +=

–4 40

– – –1 22

0 –3 18

6

–4 40

–4 6 40

16

d t

d t

t

t

d t

d

d



ANSWERS

ANSWERS

1. B y = 4x – 18

2. C No, because the slope is the same and the intercepts are different.

3. a. Day TicketsNumber of Tickets 20,000 25,000 30,000 35,000 40,000 45,000

Income ($) 1,600,000 2,000,000 2,400,000 2,800,000 3,200,000 3,600,000

b. Income = $80(number of tickets)

y = 80x

c.

500

0

1,000

1,500

2,000

2,500

3,000

3,500

4,000

50 60403020100 x

yDay Tickets

Number of Tickets (in 1,000s)

Inco

me

(in $

1,00

0s)

d. Full Festival TicketsNumber of Tickets 20,000 25,000 30,000 35,000 40,000 45,000

Income ($) 4,000,000 5,000,000 6,000,000 7,000,000 8,000,000 9,000,000

(continues)

LESSON 7: COMPARING LINEAR FUNCTIONS



ANSWERSLESSON 7: COMPARING LINEAR FUNCTIONS

(continued)

e. Income = $200(number of tickets)

y = 200x

f.

500

0

1,000

1,500

2,000

2,500

3,000

3,500

4,500

4,000

50 60403020100 x

yDay Tickets and Full Festival Tickets

Number of Tickets (in 1,000s)

Inco

me

(in $

1,00

0s)

y = 80xDay tickets

Full festivaltickets

y = 200x

g. Day ticket rate of change: $80 per ticket

Full festival ticket rate of change: $200 per ticket

Challenge Problem

4. Since you make the most profit from day tickets you would want to sell a total of 60,000 day tickets, which means 20,000 people per day. Ideally the rest of the space will be filled with people who have full festival tickets. The maximum capacity is 50,000 people, so if you sell 30,000 full festival tickets the festival will be filled each day. Total earnings from ticket sales would be $10,800,000.

Total = ($80 • 60,000) + ($200 • 30,000)

= $4,800,000 + $6,000,000

= $10,800,000



ANSWERS

ANSWERS

1. A 1

2. A The graph decreases from x = 0 to x = 6.

3. From these two points you can determine that the rate of change is –34

. You know the y-intercept, since it is given.

a. The value will reach 0 at the point (16, 0).

b. The equation for this linear relationship can be written y = –34

x + 12.

4. a. The range of the function is the y-values interval of [–11, 49].

b. f(x) = 6x – 11

Substitute 6 and 26 for x and y. 26 = 6(6) – 11 26 ≠ 25 Contradiction!

The point (6, 26) does not fit this function.

c. g(x) = 6x – 10

5. Using the times given you can determine that the rate of the water level dropping is 1.5 ft/hr. The equation for this function can be written:

f(t) = 12 – 1.5t t = number of hours after 6 a.m.

The equation for the process of draining the tank is:

===

0 12 – 1.5

1.5 12

8 hr

t

t

t

Since the tank starts being drained at 6 a.m. it will be completely drained at 2 p.m.

Challenge Problem

6. a. From 1995 you can create an equation that models the amount of cheese based on time (in years), where t = 0 means 1995. f(t) = 700,000,000 + 36,000,000t

b. The year 2013 is 18 years after 1995. So substitute 18 for t.

f(18) = 700,000,000 + 36,000,000(18) = 700,000,000 + 648,000,000 = 1,348,000,000 kg of cheese

If this growth model still holds in 2013, the Dutch production of cheese for 2013 would be 1,348,000,000 kg.

LESSON 8: GRAPHS AND FORMULAS



ANSWERS

ANSWERS

1. A Season pass

2. D y = 10x + 50

3. a. f (t) = 10t, or d = 10t

b. Time t is positive in this situation. The domain is t ≥ 0.

c. f (3) = 10(3) = 30 mi

d. Answers will vary.

e.

Time (hr)

0

10

30

40

50

2 4 6 8 10

Dis

tanc

e (m

i) 60

70

80

20

0

d

t1 3 5 7 9

4. a. g(d) = 0.1d, or t = 1

10d

b. Distance d is positive in this situation. The domain is d ≥ 0.

c. g(30) = 0.1(30) = 3 hr

d. Answers will vary.

e.

Distance (mi)

080

Tim

e (h

r)

10

0

t

d10 20 30 40 50 60 70

123456789

LESSON 9: MODEL SITUATIONS



ANSWERSLESSON 9: MODEL SITUATIONS

5. a. The total area of the path is 3x2 + 24x. The rough sketch shows how the area was calculated.

7x 5x

7x

5xx2

x2 x2

b. A(x) = 3x2 + 24x Substitute the value 1 for x:

A(1) = 3(1)2 + 24(1)

A(1) = 27

c. Substitute the value 1.5 for x:

A(1.5) = 3(1.5)2 + 24(1.5)

A(1.5) = 42.75



ANSWERSLESSON 9: MODEL SITUATIONS

6. a. Jack charges $50 to come to your house and then $50 per hour. Marta charges $65 to come to your house and then $40 per hour. Marta has a higher initial cost but a lower hourly rate.

b. Both equations are in terms of t (in hours) and are for the total cost of the plumber. The equation for Jack: J(t) = 50 + 50t The equation for Marta: M(t) = 65 + 40t

c. A good way to analyze these two functions is to visualize them by graphing them on the same coordinate system.

50

0

100

150

200

210 43 x

y

Cos

t ($

)

Hours on the Job

Plumber Cost

JackMarta

d. The graph shows that the costs are exactly the same at 1.5 hr. So, if the visit will take less than 1.5 hr you should hire Jack, but if the visit will take longer than 1.5 hr you should hire Marta.



ANSWERS

ANSWERS

1. a. f (x) = –1,500x + 19,000 or y = 19,000 – 1,500x

b. The y-intercept is (0, 19,000).

c. Domain of f : 0 ≤ t ≤ 3

d. g(3) = 14,500 – 2,500(3) = 14,500 – 7,500 = 7,000 ft

e. Domain of g : 3 ≤ t < 6

f. 6 hr

g. 7,000 19,000–

–6 0 =

–12,0006

= –2,000 ft/hr

h. h(x) = –2,000x + 19,000 or y = 19,000 – 2,000x

i. h(4) = 19,000 – 2,000(4) = 19,000 – 8,000 = 11,000 ft

2. a. Miami: (x is measured miles)

M(x) = 2.50 for x [0, 16

]

M(x) = 2.50 + [0.40 • 6(x – 16

)] for x >16

New York: N(x) = 2.50 for x [0, 0.2] N(x) = 2.50 + 2[5(x–0.2)] for x > 0.2

b. Taxi Fares

Distance (mi)

Cos

t ($

)

0.51.0

0

1.52.02.53.03.54.04.55.05.56.06.5

1.00.80.60.40.20 x

y

New York

Miami

(continues)

LESSON 11: MORE MODELING



ANSWERSLESSON 11: MORE MODELING

(continued)

c. If you are going less than 16

mi there is no difference.

If you are going between 16

and 15

mi then New York is slightly cheaper.

If you are going more than 15

mi then Miami is cheaper.

As you go farther and farther Miami is significantly cheaper.

3. a.

b. The two lines do not intersect on this domain.

c. The range of f(x) is the set of y-values [–5, 35].

The range of g(x) is the set of y-values [–38, 32].

10

20

30

40

–30

–40

–20

–10

5–5 x

y

f(x) = 4x + 15 g(x) = 7x – 3



ANSWERSLESSON 11: MORE MODELING

Challenge Problem

4. f(x) and g(x) intersect at (–6.193, –8.135)

2

4

–12

–10

–8

–6

–4

–28642–2–4–6–8–10 x

y

f(x) = 3.8x + 15.4

g(x) = 0.7x – 3.8



ANSWERS

ANSWERS

1. D 2 sec

2. D You may need more than three coordinate pairs.

3. Both graphs have the same start and end points on this domain. The difference is that the first graph is slightly curved while the second is a straight line.

4. From the graph it looks like the car has gone almost 55 m after 2 sec of braking.

5. a. f(x) is linear while g(x) is curved. On the domain from before, [0, 3.2], g(x) is higher, but after x = 3.2 then f(x) is higher.

Stopping Distances

Time (sec)

Dis

tanc

e (m

)

f(x) = 25x

g(x) = –1.5x2 + 30x

200

40

220240260

6080

100120140160180200

10 119876543210 x

y

b. g(x) will eventually stop gaining distance, which would show when the car is stopped. In fact, you can see at the edge of this graph that the car will be stopped at 10 sec.

c. The range of f(x) is the set of y-values from [0, 250].

The range of g(x) is the set of y-values from [0, 150].

LESSON 13: USING FUNCTIONS TO PREDICT



ANSWERSLESSON 13: USING FUNCTIONS TO PREDICT

Challenge Problem

6. a.

f(x) = 1.4 – 0.1x

0.5

1.0

1.5

108642 161412 x

y Herring

Years after 1989

Mill

ions

of T

ons

b. 2005 is 16 years after 1989. On the graph, x = 16 actually shows a negative value (–0.2 million) for the herring population, meaning that the fish has gone extinct!

c. From the real data you can see that the herring numbers began to increase again around 1996; the herring did not go extinct as the graph from part a showed.

Documents

FUNCTIONS - RUSD Mathrusdmath.weebly.com/uploads/1/1/1/5/11156667/g8_u7... · 2018-10-14 · Day Tickets and Full Festival Tickets Number of Tickets (in 1,000s) Income (i n $1,0 00