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D. Applications of Composition and the Difference Quotient

# D. Applications of Composition and the Difference Quotient

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Solution

363

a. Since r16002 ϭ 4300, there are currently 4300 rodents in the county. The number

of rodents and small animals is a function of w: r1w2 ϭ 10,000 Ϫ 9.5w, and the

number of wolves w is a function of human population x: w1x2 ϭ 600 Ϫ 0.02x.

To find a function for r in terms of x, we use the composition 1r ‫ ؠ‬w21x2 ϭ r3w1x24 .

r1w2 ϭ 10,000 Ϫ 9.5w

r 3w1x2 4 ϭ 10,000 Ϫ 9.53600 Ϫ 0.02x 4

ϭ 10,000 Ϫ 5700 ϩ 0.19x

r1x2 ϭ 4300 ϩ 0.19x

r depends on w

compose r with w (x )

distribute

simplify, r depends on x

b. Evaluating r(35,000) gives 10,950 rodents, showing a decline in the wolf

population will eventually cause the rodent population to flourish, with an adverse

effect on humans. In cases like these, a careful balance should be the goal.

Now try Exercises 63 and 64

You might be familiar with Galileo Galilei and his studies of gravity. According to

popular history, he demonstrated that unequal weights will fall equal distances in equal

time periods, by dropping cannonballs from the upper floors of the Leaning Tower of

Pisa. Neglecting air resistance, the distance an object falls is modeled by the function

d1t2 ϭ 16t2, where d(t) represents the distance fallen after t sec. Due to the effects of

gravity, the velocity of the object increases as it falls. In other words, the velocity or the

¢distance

average rate of change

is a nonconstant (increasing) quantity. We can analyze

¢time

this rate of change using the difference quotient.

EXAMPLE 11

Applying the Difference Quotient in Context

A construction worker drops a heavy wrench from atop a

girder of a new skyscraper. Use the function d1t2 ϭ 16t2 to

a. Compute the distance the wrench has fallen after

2 sec and after 7 sec.

b. Find a formula for the velocity of the wrench (average

rate of change in distance per unit time).

c. Use the formula to find the rate of change in the

intervals [2, 2.01] and [7, 7.01].

d. Graph the function and the secant lines representing

the average rate of change. Comment on what you

notice.

Solution

a. Substituting t ϭ 2 and t ϭ 7 in the given function

yields

d122 ϭ 16122 2

ϭ 16142

ϭ 64

d172 ϭ 16172 2

ϭ 161492

ϭ 784

evaluate d 1t 2 ϭ 16t 2

square input

multiply

After 2 sec, the wrench has fallen 64 ft; after 7 sec, the wrench has fallen 784 ft.

b. For d1t2 ϭ 16t2, d1t ϩ h2 ϭ 161t ϩ h2 2, which we compute separately.

d1t ϩ h2 ϭ 161t ϩ h2 2

ϭ 161t2 ϩ 2th ϩ h2 2

ϭ 16t2 ϩ 32th ϩ 16h2

substitute t ϩ h for t

square binomial

distribute 16

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Using this result in the difference quotient yields

d1t ϩ h2 Ϫ d1t2

h

ϭ

116t2 ϩ 32th ϩ 16h2 2 Ϫ 16t2

h

2

16t ϩ 32th ϩ 16h2 Ϫ 16t2

ϭ

h

32th ϩ 16h2

ϭ

h

h132t ϩ 16h2

ϭ

h

ϭ 32t ϩ 16h

substitute into the difference quotient

eliminate parentheses

combine like terms

factor out h and simplify

result

For any number of seconds t and h a small increment of time thereafter, the

32t ؉ 16h

¢distance

‫؍‬

velocity of the wrench is modeled by

.

¢time

1

c. For the interval 3t, t ϩ h 4 ϭ 3 2, 2.01 4, t ϭ 2 and h ϭ 0.01:

32122 ϩ 1610.012

¢distance

ϭ

¢time

1

ϭ 64 ϩ 0.16 ϭ 64.16

substitute 2 for t and 0.01 for h

Two seconds after being dropped, the velocity of the wrench is close to

64.16 ft/sec (44 mph). For the interval 3 t, t ϩ h 4 ϭ 3 7, 7.014 , t ϭ 7 and h ϭ 0.01:

32172 ϩ 1610.012

¢distance

ϭ

substitute 7 for t and 0.01 for h

¢time

1

ϭ 224 ϩ 0.16 ϭ 224.16

Seven seconds after being dropped, the velocity of the wrench is

approximately 224.16 ft/sec (about 153 mph).

d.

y

1000

Distance fallen (ft)

800

600

400

200

0

1

2

3

4

5

6

7

8

9

10

x

Time in seconds

D. You’ve just seen how

we can apply the composition

of functions and the difference

quotient in context

The velocity increases with time, as indicated by the steepness of each

secant line.

Now try Exercises 65 through 68

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365

3.6 EXERCISES

CONCEPTS AND VOCABULARY

Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.

1. When evaluating functions, if the input value is a

function itself, the process is called a

of

functions.

3. For functions f and g, the domain of 1 f ‫ ؠ‬g21x2 is the

set of all x in the

of g, such that

is in the domain of f.

5. Discuss/Explain how and why using the the

difference quotient differs from using the average

rate of change formula.

2. The notation 1 f ‫ ؠ‬g21x2 indicates that g(x) is the input

value for f (x), which is written

.

4. The average rate of change formula becomes the

quotient by substituting

for

for x1.

x2 and

6. Discuss/Explain how the domain of 1 f ‫ ؠ‬g21x2

is determined, given f 1x2 ϭ 12x ϩ 7 and

2x

g1x2 ϭ

.

xϪ1

7. Given f 1x2 ϭ x2 Ϫ 5x Ϫ 14, find f 1Ϫ22, f (7),

f (2a), and f 1a Ϫ 22 .

8. Given g1x2 ϭ x3 Ϫ 9x, find g1Ϫ32, g122, g(3t), and

g1t ϩ 12 .

For each pair of functions below, find (a) h(x) ϭ ( f ‫ ؠ‬g)(x)

and (b) H(x) ϭ ( g ‫ ؠ‬f )(x), and (c) determine the domain

of each result.

9. f 1x2 ϭ 1x ϩ 3 and g1x2 ϭ 2x Ϫ 5

10. f 1x2 ϭ x ϩ 3 and g1x2 ϭ 29 Ϫ x

11. f 1x2 ϭ 1x Ϫ 3 and g1x2 ϭ 3x ϩ 4

12. f 1x2 ϭ 1x ϩ 5 and g1x2 ϭ 4x Ϫ 1

21. f 1x2 ϭ 1x ϩ 82 2, g1x2 ϭ

22. f 1x2 ϭ

72

xϪ5

1

1

, g1x2 ϭ

2

2x Ϫ 1

x

23. f 1x2 ϭ 24 Ϫ 3x, g1x2 ϭ x2 Ϫ 9

24. f 1x2 ϭ

7

, g1x2 ϭ x2 Ϫ 11

xϩ2

For the functions f(x) and g(x) given, analyze the

domain of (a) ( f ‫ ؠ‬g)(x) and (b) ( g ‫ ؠ‬f )(x), then (c) find

the actual compositions and comment.

25. f 1x2 ϭ

2x

5

and g1x2 ϭ

x

xϩ3

26. f 1x2 ϭ

Ϫ3

x

and g1x2 ϭ

x

xϪ2

16. f 1x2 ϭ x2 Ϫ 4x ϩ 2 and g1x2 ϭ x Ϫ 2

27. f 1x2 ϭ

1

4

and g1x2 ϭ

x

xϪ5

18. f 1x2 ϭ Ϳ x Ϫ 2 Ϳ and g1x2 ϭ 3x Ϫ 5

28. f 1x2 ϭ

3

1

and g1x2 ϭ

x

xϪ2

13. f 1x2 ϭ x2 Ϫ 3x and g1x2 ϭ x ϩ 2

14. f 1x2 ϭ 2x2 Ϫ 1 and g1x2 ϭ 3x ϩ 2

15. f 1x2 ϭ x ϩ x Ϫ 4 and g1x2 ϭ x ϩ 3

2

17. f 1x2 ϭ Ϳ x Ϳ Ϫ 5 and g1x2 ϭ Ϫ3x ϩ 1

For the functions f and g given, h(x) ϭ ( f ‫ ؠ‬g)(x). Use a

1

calculator to evaluate h(Ϫ3), h( 22), ha b, and h(5). If

2

an error message is received, explain why.

19. f 1x2 ϭ x ϩ 3x Ϫ 4, g1x2 ϭ x ϩ 1

2

20. f 1x2 ϭ Ϫx2 Ϫ 15x, g1x2 ϭ x Ϫ 2

29. For f 1x2 ϭ x2 Ϫ 8, g1x2 ϭ x ϩ 2, and

h1x2 ϭ 1 f ‫ ؠ‬g21x2, find h(5) in two ways:

a. 1 f ‫ ؠ‬g2152

b. f [g(5)]

30. For p1x2 ϭ x2 Ϫ 8, q1x2 ϭ x ϩ 2, and

H1x2 ϭ 1p ‫ ؠ‬q21x2, find H1Ϫ22 in two ways:

a. 1p ‫ ؠ‬q21Ϫ22

b. p 3q1Ϫ22 4

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31. For h1x2 ϭ 1 1x Ϫ 2 ϩ 12 3 Ϫ 5, find two

functions f and g such that 1f ‫ ؠ‬g2 1x2 ϭ h1x2 .

38. Given f 1x2 ϭ 1x, g1x2 ϭ

1

, (a) state

x Ϫ 2x Ϫ 3

the domain of f and g, then (b) use a graphing

calculator to study the graph of h1x2 ϭ 1 f ‫ ؠ‬g21x2 .

Finally, (c) algebraically determine the domain of

h, and reconcile it with the graph.

3 2

32. For H1x2 ϭ 2

x Ϫ 5 ϩ 2, find two functions

p and q such that 1p ‫ ؠ‬q2 1x2 ϭ H1x2 .

33. Given f 1x2 ϭ 2x Ϫ 1, g1x2 ϭ x2 Ϫ 1, and

h1x2 ϭ x ϩ 4, find p1x2 ϭ f 3 g1 3 h1x2 4 2 4 and

q1x2 ϭ g3 f 1 3 h1x2 4 2 4 .

xϪ3

, find

2

(a) 1 f ‫ ؠ‬f 2 1x2 , (b) 1g ‫ ؠ‬g21x2 , (c) 1 f ‫ ؠ‬g21x2 , and

(d) 1g ‫ ؠ‬f 2 1x2 .

the given graph to find the

result of the operations

indicated.

Note f 1Ϫ42 ϭ 5,

g1Ϫ42 ϭ Ϫ1, and

so on.

a. 1 f ‫ ؠ‬g21Ϫ42

b. 1 f ‫ ؠ‬g2112

c. 1 f ‫ ؠ‬g2142

e. 1 f ‫ ؠ‬g21Ϫ22

g. 1 f ‫ ؠ‬g2162

Exercise 35

y

f(x)

Note p1Ϫ12 ϭ Ϫ2,

q152 ϭ 6, and so on.

a. 1p ‫ ؠ‬q21Ϫ42

b. 1p ‫ ؠ‬q2112

c. 1p ‫ ؠ‬q2142

e. 1p ‫ ؠ‬q21Ϫ22

g. 1q ‫ ؠ‬q21Ϫ12

6

g(x)

Ϫ4

8 x

Ϫ4

d. 1 f ‫ ؠ‬g2102

f. 1g ‫ ؠ‬f 2122

h. 1g ‫ ؠ‬f 2142

40. g1x2 ϭ 4x ϩ 1

41. j1x2 ϭ x2 ϩ 3

42. p1x2 ϭ x2 Ϫ 2

43. q1x2 ϭ x2 ϩ 2x Ϫ 3

44. r1x2 ϭ x2 Ϫ 5x ϩ 2

45. f 1x2 ϭ

46. g1x2 ϭ

2

x

Ϫ3

x

Use the difference quotient to find: (a) a rate of change

formula for the functions given and (b)/(c) calculate the

rate of change in the intervals shown. Then (d) sketch

the graph of each function along with the secant lines

and comment on what you notice.

48. j1x2 ϭ x2 Ϫ 6x

[Ϫ3.0, Ϫ2.9], [0.50, 0.51]

[1.9, 2.0], [5.0, 5.01]

49. g1x2 ϭ x ϩ 1

[Ϫ2.1, Ϫ2], [0.40, 0.41]

3

y

6

39. f 1x2 ϭ 2x Ϫ 3

47. g1x2 ϭ x2 ϩ 2x

Exercise 36

the given graph to find the

result of the operations

indicated.

2

Compute and simplify the difference quotient

f 1x ؉ h2 ؊ f 1x2

for each function given.

h

34. Given f 1x2 ϭ 2x ϩ 3 and g1x2 ϭ

50. v1x2 ϭ 2x (Hint: Rationalize the numerator.)

p(x)

[1, 1.1], [4, 4.1]

Ϫ4

4

Ϫ4

8 x

q(x)

d. 1p ‫ ؠ‬q2102

f. 1q ‫ ؠ‬p2122

h. 1p ‫ ؠ‬p2172

Ϫ3

, (a) state

xϩ2

the domain of f and g, then (b) use a graphing

calculator to study the graph of h1x2 ϭ 1 f ‫ ؠ‬g21x2 .

Finally, (c) algebraically determine the domain of

h, and reconcile it with the graph.

37. Given f 1x2 ϭ 3 2x ϩ 1, g1x2 ϭ

3–86

CHAPTER 3 Quadratic Functions and Operations on Functions

Use the difference quotient to find a rate of change

formula for the functions given, then calculate the rate

of change for the intervals indicated. Comment on how

the rate of change in each interval corresponds to the

graph of the function.

1

52. f 1x2 ϭ x2 Ϫ 4x

x2

30.00, 0.014 , 33.00, 3.014

30.50, 0.514 , 31.50, 1.514

51. j1x2 ϭ

53. g1x2 ϭ x3 ϩ 1

54. r1x2 ϭ 1x

3Ϫ2.01, Ϫ2.004, 30.40, 0.414

31.00, 1.014, 34.00, 4.014

WORKING WITH FORMULAS

55. Transformations via composition: For

f 1x2 ϭ x2 ϩ 4x ϩ 3 and g1x2 ϭ x Ϫ 2, (a) show that

h1x2 ϭ 1 f ‫ ؠ‬g21x2 ϭ x2 Ϫ 1, then (b) verify the graph

of h is the same as that of f, shifted 2 units to the right.

56. Compound annual growth: A1r2 ‫ ؍‬P11 ؉ r2 t

The amount of money A in a savings account t yr

after an initial investment of P dollars depends on

the interest rate r. If \$1000 is invested for 5 yr, find

f (r) and g(r) such that A1r2 ϭ 1 f ‫ ؠ‬g21r2.

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APPLICATIONS

57. International shoe sizes: Peering inside her

athletic shoes, Morgan notes the following shoe

sizes: US 8.5, UK 6, EUR 40. The function that

relates the U.S. sizes to the European (EUR) sizes

is g1x2 ϭ 2x ϩ 23, where x represents the U.S.

size and g(x) represents the EUR size. The function

that relates European sizes to sizes in the United

Kingdom (UK) is f 1x2 ϭ 0.5x Ϫ 14, where x

represents the EUR size and f (x) represents the UK

size. Find the function h(x) that relates the U.S.

measurement directly to the UK measurement by

finding h1x2 ϭ 1 f ‫ ؠ‬g2 1x2. Find the UK size for a

shoe that has a U.S. size of 13.

58. Currency conversion: On a trip to Europe, Megan

had to convert American dollars to euros using the

function E1x2 ϭ 1.12x, where x represents the

number of dollars and E(x) is the equivalent

number of euros. Later, she converts her euros to

Japanese yen using the function Y1x2 ϭ 1061x,

where x represents the number of euros and Y(x)

represents the equivalent number of yen.

(a) Convert 100 U.S. dollars to euros. (b) Convert

the answer from part (a) into Japanese yen.

(c) Express yen as a function of dollars by finding

M1x2 ϭ 1Y ‫ ؠ‬E21x2, then use M(x) to convert \$100

directly to yen. Do parts (b) and (c) agree?

Source: 2005 World Almanac, p. 231

59. Currency conversion: While traveling in the Far

East, Timi must convert U.S. dollars to Thai baht

using the function T1x2 ϭ 41.6x, where x

represents the number of dollars and T(x) is the

equivalent number of baht. Later she needs to

convert her baht to Malaysian ringgit using the

function R1x2 ϭ 10.9x. (a) Convert \$100 to baht.

(b) Convert the result from part (a) to ringgit.

(c) Express ringgit as a function of dollars using

M1x2 ϭ 1R ‫ ؠ‬T21x2, then use M(x) to convert \$100

to ringgit directly. Do parts (b) and (c) agree?

Source: 2005 World Almanac, p. 231

60. Spread of a fire: Due to a lightning strike, a forest

fire begins to burn and is spreading outward in a

shape that is roughly circular. The radius of the

circle is modeled by the function r1t2 ϭ 2t, where t

is the time in minutes and r is measured in meters.

(a) Write a function for the area burned by the fire

directly as a function of t by computing 1A ‫ ؠ‬r2 1t2 .

(b) Find the area of the circular burn after 60 min.

61. Radius of a ripple: As Mark drops firecrackers

into a lake one 4th of July, each “pop” caused a

circular ripple that expanded with time. The

radius of the circle is a function of time t.

Suppose the function is r1t2 ϭ 3t, where t is in

seconds and r is in feet. (a) Find the radius of the

circle after 2 sec. (b) Find the area of the circle

after 2 sec. (c) Express the area as a function of

time by finding A1t2 ϭ 1A ‫ ؠ‬r21t2 and use A(t) to

find the area of the circle after 2 sec. Do the

62. Expanding supernova: The surface area of a star

goes through an expansion phase prior to going

supernova. As the star begins expanding, the

radius becomes a function of time. Suppose this

function is r 1t2 ϭ 1.05t, where t is in days and r(t)

is in gigameters (Gm). (a) Find the radius of the

star two days after the expansion phase begins.

(b) Find the surface area after two days. (c)

Express the surface area as a function of time by

finding h1t2 ϭ 1S ‫ ؠ‬r2 1t2, then use h(t) to compute

the surface area after two days directly. Do the

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63. Composition and dependent relationships: In

the wild, the balance of nature is often very fragile,

with any sudden changes causing dramatic and

unforeseen changes. With a huge increase in

population and tourism near an African wildlife

preserve, the number of lions is decreasing due to

loss of habitat and a disruption in normal daily

movements. This is causing a related increase in

the hyena population, as the lion is one of the

hyena’s only natural predators. If this increase

remains unchecked, animals lower in the food

chain will suffer. If the lion population L depends

on the increase in human population x according

to the formula L1x2 ϭ 500 Ϫ 0.015x, and the

hyena population depends on the lion population

as modeled by the formula H1L2 ϭ 650 Ϫ 0.5L,

(a) what is the current lion population 1x ϭ 02

and hyena population? (b) Use a composition to

find a function modeling how the hyena population

relates directly to the number of humans, and use

the function to estimate the number of hyenas in

the area if the human population grows by 16,000.

(c) If the administrators of the preserve consider a

population of 625 hyenas as “extremely detrimental,”

at what point should the human population be

capped?

64. Composition and dependent relationships: The

recent opening of a landfill in the area has caused

the raccoon population to flourish, with an

adverse effect on the number of purple martins.

Wildlife specialists believe the population of

martins p will decrease as the raccoon population

r grows. Further, since mosquitoes are the primary

diet of purple martins, the mosquito population

m is likewise affected. If the first relationship is

modeled by the function p1r2 ϭ 750 Ϫ 3.75r and

the second by m1p2 ϭ 50,000 Ϫ 45p, (a) what is

the current number of purple martins 1r ϭ 02 and

mosquitoes? (b) Use a composition to find a

function modeling how the raccoon population

relates directly to the number of mosquitoes,

and use the function to estimate the number of

mosquitoes in the area if the raccoon population

3–88

grows by 50. (c) If the health department considers

36,500 mosquitoes to be a “dangerous level,”

what increase in the raccoon population will

65. Distance to the horizon: The distance that a

person can see depends on how high they’re

standing above level ground. On a clear day,

the distance is approximated by the function

d1h2 ϭ 1.2 1h, where d(h) represents the viewing

distance (in miles) at height h (in feet). Use

the difference quotient to find the average rate

of change in the intervals (a) [9, 9.01] and

(b) [225, 225.01]. Then (c) graph the function

along with the lines representing the average

rates of change and comment on what you

notice.

66. Projector lenses: A special magnifying lens is

crafted and installed in an overhead projector.

When the projector is x ft from the screen, the

size P(x) of the projected image is x 2. Use the

difference quotient to find the average rate of

change for P1x2 ϭ x 2 in the intervals (a) [1, 1.01]

and (b) [4, 4.01]. Then (c) graph the function

along with the lines representing the average

rates of change and comment on what you

notice.

67. Fortune and fame: Over the years there have

been a large number of what we know as “one

hit wonders,” persons or groups that published a

memorable or timeless song, but who were

unable to repeat the feat. In some cases, their

fame might be modeled by a quadratic function

as their popularity rose to a maximum, then

faded with time. Suppose the song She’s on Her

Way by Helyn Wheels rode to the top of the

charts in January of 1988, with demand for the

song modeled by d1t2 ϭ Ϫ2t2 ϩ 27t. Here, d(t)

represents the demand in 1000s for month

t 1t ϭ 1 S Jan2 . (a) How many times faster was

the demand growing in March (shortly after

the release) than in June? Use the difference

quotient and the intervals [3, 3.01] for March

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and [6, 6.01] for June. (b) Determine the month

that demand reached its peak using a graphing

calculator. (c) Was the demand increasing or

decreasing in the month of August? At what

rate?

68. Velocity and fuel economy: It has long been

known that cars and trucks are more fuel efficient

at certain speeds, which is why President Richard

Nixon lowered the speed limit on all federal

highways to 55 mph during the oil embargo of

1974. For heavier and less fuel-efficient vehicles,

the miles per gallon for certain speeds can be

modeled by the function m1s2 ϭ Ϫ0.01s2 ϩ s,

where m(s) represents the mileage (in miles per

gallon) at speed s 10 6 s Յ 802 . (a) Use the

difference quotient to find how many times

faster fuel efficiency is growing near s ϭ 30 mph

than near s ϭ 45 mph. Use the intervals [30, 30.1]

and [45, 45.1]. (b) Use a graphing calculator

to determine the speed(s) that maximizes fuel

efficiency for this vehicle. How many miles

per gallon are achieved? (c) Is fuel efficiency

increasing or decreasing at 70 mph? At what

rate?

EXTENDING THE CONCEPT

3

69. Given f 1x2 ϭ x3 ϩ 2 and g1x2 ϭ 2

x Ϫ 2, graph

each function on the same axes by plotting the

points that correspond to integer inputs for

x ʦ 3 Ϫ3, 3 4. Do you notice anything? Next, find

h1x2 ϭ 1 f ‫ ؠ‬g21x2 and H1x2 ϭ 1g ‫ ؠ‬f 2 1x2. What

happened? Look closely at the functions f and g to

see how they are related. Can you come up with two

additional functions where the same thing occurs?

70. Given f 1x2 ϭ

1

, g1x2 ϭ 1x ϩ 1, and

x Ϫ4

h1x2 ϭ 1 f ‫ ؠ‬g21x2 , (a) find the new function rule for

h and (b) determine the implied domain of h. Does

this implied domain include x ϭ 2, x ϭ Ϫ2, and

x ϭ Ϫ3 as valid inputs? (c) Determine the actual

domain for h1x2 ϭ 1 f ‫ ؠ‬g21x2 and discuss the result.

369

2

71. Consider the functions f 1x2 ϭ

k

k

and g1x2 ϭ 2 .

x

x

Both graphs appear similar in Quadrant I and both

may “fit” a scatterplot fairly well, but there is a big

difference between them — they decrease as x gets

larger, but they decrease at very different rates.

(a) Assume k ϭ 1 and use the ideas from this

section to compute the rates of change for f and g

for the interval from x ϭ 0.5 to x ϭ 0.51. Were you

surprised? (b) In the interval x ϭ 0.8 to x ϭ 0.81,

will the rate of decrease for each function be

greater or less than in the interval x ϭ 0.5 to

x ϭ 0.51? Why?

72. (3.1) Find the sum and product of the complex

numbers 2 ϩ 3i and 2 Ϫ 3i.

74. (3.2) Use the quadratic formula to solve

2x2 Ϫ 3x ϩ 4 ϭ 0.

73. (2.2) Draw a sketch of the functions from memory.

3

(a) f 1x2 ϭ 1x,

(b) g1x2 ϭ 2x, and

(c) h1x2 ϭ ͿxͿ

75. (1.4) Find an equation of the line perpendicular to

Ϫ2x ϩ 3y ϭ 9, that also goes through the origin.

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CHAPTER 3 Quadratic Functions and Operations on Functions

MAKING CONNECTIONS

Making Connections: Graphically, Symbolically, Numerically, and Verbally

Eight graphs (a) through (h) are given. Match the characteristics shown in 1 through 16 to one of the eight graphs.

y

(a)

Ϫ5

y

(b)

5

Ϫ5

5 x

y

Ϫ5

Ϫ5

5 x

5 x

Ϫ5

1. ____ 1x ϩ 12 2 ϩ 1y Ϫ 22 2 ϭ 4

2

2. ____ y ϭ Ϫ x ϩ 1

5

y

(h)

5

Ϫ5

5 x

5

Ϫ5

Ϫ5

9. ____ center 11, Ϫ22, radius ϭ 3

10. ____ vertex (2, 5), y-intercept (0, 1)

11. ____ y ϭ

5. ____ m 6 0, b 7 0

13. ____ axis of symmetry x ϭ 2,

opens downward

6. ____ m 7 0, b 6 0

14. ____ 4x Ϫ 3y ϭ 3

7. ____ y ϭ Ϫx ϩ 4

2

8. ____ y ϭ 31x Ϫ 22 2 Ϫ 5

5 x

Ϫ5

3. ____ f 1x2T for x ʦ 1Ϫq, 22 ,

f 1x2c for x ʦ 12, q2

4. ____ f 1x2c for x ʦ 1Ϫq, 22 ,

f 1x2T for x ʦ 12, q2

5 x

Ϫ5

y

(g)

5

Ϫ5

5 x

5

Ϫ5

y

(f)

5

Ϫ5

Ϫ5

5 x

y

(d)

5

Ϫ5

Ϫ5

(e)

y

(c)

5

1

1x Ϫ 12 2 Ϫ 3

3

12. ____ 1x Ϫ 12 2 ϩ 1y ϩ 22 2 ϭ 9

15. ____ f 1x2 6 0 for x ʦ 1Ϫq,Ϫ22 ´ 12, q 2 ,

f 1x2 Ն 0 for x ʦ 3Ϫ2, 2 4

16. ____ f 1Ϫ22 ϭ 0, f 112 ϭ Ϫ3

SUMMARY AND CONCEPT REVIEW

SECTION 3.1

Complex Numbers

KEY CONCEPTS

• The italicized i represents the number whose square is Ϫ1. This means i 2 ϭ Ϫ1 and i ϭ 1Ϫ1.

• Larger powers of i can be simplified using i 4 ϭ 1.

• For k 7 0, 1Ϫk ϭ i1k and we say the expression has been written in terms of i.

• The standard form of a complex number is a ϩ bi, where a is the real number part and bi is the imaginary part.

• To add or subtract complex numbers, combine the like terms.

• For any complex number a ϩ bi, its complex conjugate is a Ϫ bi.

• The product of a complex number and its conjugate is a real number.

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371

• The commutative, associative, and distributive properties also apply to complex numbers and are used to perform

basic operations.

• To multiply complex numbers, use the F-O-I-L method and simplify.

• To find a quotient of complex numbers, multiply the numerator and denominator by the conjugate of the denominator.

EXERCISES

Simplify each expression and write the result in standard form.

1. 1Ϫ72

2. 6 1Ϫ48

4. 13 1Ϫ6

5. i57

3.

Perform the operation indicated and write the result in standard form.

5i

6. 15 ϩ 2i2 2

7.

1 Ϫ 2i

9. 12 ϩ 3i212 Ϫ 3i2

10. 4i1Ϫ3 ϩ 5i2

Ϫ10 ϩ 1Ϫ50

5

8. 1Ϫ3 ϩ 5i2 Ϫ 12 Ϫ 2i2

Use substitution to show the given complex number and its conjugate are solutions to the equation shown.

11. x2 Ϫ 9 ϭ Ϫ34; x ϭ 5i

SECTION 3.2

12. x2 Ϫ 4x ϩ 9 ϭ 0; x ϭ 2 ϩ i 25

KEY CONCEPTS

• The standard form of a quadratic equation is ax2 ϩ bx ϩ c ϭ 0, where a, b, and c are real numbers and a 0. In

words, we say the equation is written in decreasing order of degree and set equal to zero.

• A quadratic function is one that can be written as f 1x2 ϭ ax2 ϩ bx ϩ c, where a, b, and c are real numbers and a 0.

• The following four statements are equivalent: (1) x ϭ r is a solution of f 1x2 ϭ 0, (2) r is a zero of f(x), (3) (r, 0) is

an x-intercept of y ϭ f 1x2 , and (4) 1x Ϫ r2 is a factor of f (x).

• The square root property of equality states that if X2 ϭ k, where k Ն 0, then X ϭ 1k or X ϭ Ϫ 1k.

• Quadratic equations can also be solved by completing the square, or using the quadratic formula.

• If the discriminant b2 Ϫ 4ac ϭ 0, the equation has one real (repeated) root. If b2 Ϫ 4ac 7 0, the equation has

two real roots; and if b2 Ϫ 4ac 6 0, the equation has two nonreal roots.

• Quadratic inequalities can be solved using the zeroes of the function and either an understanding of quadratic

graphs or mid-interval test values.

EXERCISES

13. Solve by factoring.

a. x2 Ϫ 3x Ϫ 10 ϭ 0

b. 2x2 Ϫ 50 ϭ 0

c. 3x2 Ϫ 15 ϭ 4x

d. x3 Ϫ 3x2 ϭ 4x Ϫ 12

14. Solve using the square root property of equality.

a. x2 Ϫ 9 ϭ 0

b. 21x Ϫ 22 2 ϩ 1 ϭ 11

c. 3x2 ϩ 15 ϭ 0

d. Ϫ2x2 ϩ 4 ϭ Ϫ46

15. Solve by completing the square. Give real number solutions in exact and approximate form.

a. x2 ϩ 2x ϭ 15

b. x2 ϩ 6x ϭ 16

c. Ϫ4x ϩ 2x2 ϭ 3

d. 3x2 Ϫ 7x ϭ Ϫ2

16. Solve using the quadratic formula. Give solutions in both exact and approximate form.

a. x2 Ϫ 4x ϭ Ϫ9

b. 4x2 ϩ 7 ϭ 12x

c. 2x2 Ϫ 6x ϩ 5 ϭ 0

17. Solve by locating the x-intercepts and noting the end-behavior of the graph.

a. x2 Ϫ x Ϫ 6 7 0

b. Ϫx2 ϩ 1 Ն 0

c. x2 Ϫ 2x ϩ 2 7 0

18. Solve using the interval test method.

a. x2 ϩ 3x Յ 4

b. x2 7 20 Ϫ x

c. x2 ϩ 4x ϩ 4 Յ 0

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3–92

Solve the following quadratic applications. For 19 and 20, recall the height of a projectile is modeled by

h ϭ Ϫ16t 2 ϩ v0 t ϩ k.

19. A projectile is fired upward from ground level with an initial velocity of 96 ft/sec. (a) To the nearest tenth of a

second, how long until the object first reaches a height of 100 ft? (b) How long until the object is again at 100 ft?

(c) How many seconds until it returns to the ground?

20. A person throws a rock upward from the top of an 80-ft cliff with an initial velocity of 64 ft/sec. (a) To the nearest

tenth of a second, how long until the object is 120 ft high? (b) How long until the object is again at 120 ft?

(c) How many seconds until the object hits the ground at the base of the cliff?

SECTION 3.3

KEY CONCEPTS

• The graph of a quadratic function is a parabola. Parabolas have three distinctive features: (1) like end-behavior on

the left and right, (2) an axis of symmetry, (3) a highest or lowest point called the vertex.

• By completing the square, f 1x2 ϭ ax2 ϩ bx ϩ c can be written as the transformation f 1x2 ϭ a1x ϩ h2 2 Ϯ k, and

graphed using transformations of y ϭ x2.

• For a quadratic function in the standard form y ϭ ax2 ϩ bx ϩ c,

• End-behavior: graph opens upward if a 7 0, opens downward if a 6 0

• Zeroes/x-intercepts (if they exist): substitute 0 for y and solve for x

• y-intercept: substitute 0 for x S 10, c2

Ϫb

Ϫb

,k ϭ fa

b

• Vertex: (h, k), where h ϭ

2a

2a

• Maximum value: If the parabola opens downward, y ϭ k is the maximum value of f.

• Minimum value: If the parabola opens upward, y ϭ k is the minimum value of f.

• Line of symmetry: x ϭ h is the line of symmetry. If 1h ϩ c, y2 is on the graph, then 1h Ϫ c, y2 is also on the graph.

EXERCISES

Graph p(x) and f (x) by completing the square and using transformations of the parent function. Graph g(x) and h(x)

using the vertex formula and y-intercept. Find the x-intercepts (if they exist) for all functions.

21. p1x2 ϭ x2 Ϫ 6x

22. f 1x2 ϭ x2 ϩ 8x ϩ 15

23. g1x2 ϭ Ϫx2 ϩ 4x Ϫ 5

24. h1x2 ϭ 4x2 Ϫ 12x ϩ 3

25. Height of a superball: A teenager tries to see how high she can bounce her superball by throwing it downward on

her driveway. The height of the ball (in feet) at time t (in seconds) is given by h1t2 ϭ Ϫ16t2 ϩ 96t. (a) How high

is the ball at t ϭ 0? (b) How high is the ball after 1.5 sec? (c) How long until the ball is 135 ft high? (d) What is

the maximum height attained by the ball? At what time t did this occur?

26. Theater Revenue: The manager of a large, 14-screen movie theater finds that if he charges \$2.50 per person for

the matinee, the average daily attendance is 4000 people. With every increase of 25 cents the attendance drops an

average of 200 people. (a) What admission price will bring in a revenue of \$11,250? (b) How many people will

purchase tickets at this price?

SECTION 3.4

Quadratic Models; More on Rates of Change

KEY CONCEPTS

• Regardless of the form of regression chosen, obtaining a regression equation uses these five steps: (1) clear out old

data, (2) enter new data, (3) set an appropriate window and display the data, (4) calculate the regression equation,

and (5) display the data and equation, and once satisfied the model is appropriate, apply the result.

• The choice of a nonlinear regression model often depends on many factors, particularly the context of the data,

any patterns formed by the scatterplot, some foreknowledge on how the data might be related, and/or a careful

assessment of the correlation coefficient.

• Applications of quadratic regression are generally applied when a set of data indicates a gradual decrease to some

minimum value, with a matching increase afterward, or a gradual increase to some maximum, with a matching

decrease afterward.

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Summary and Concept Review

373

• For nonlinear functions, the average rate of change gives an average value for how changes in the independent

variable cause a change in the dependent variable within a specified interval.

• The average rate of change is given by the slope of a secant line through two points (x1, y1) and (x2, y2) on the

f 1x2 2 Ϫ f 1x1 2

¢y

graph, and is computed as:

ϭ

, x2 x1.

x2 Ϫ x1

¢x

EXERCISES

Year

Amount

27. While the Internet has been with us for over 20 years, its use continues to grow at a

(2000 S 0)

(billions)

rapid pace. The data in the table gives the amount of money (in billions of dollars)

1

31

consumers spent online for retail items in selected years (amounts for 2010 through

5

84

2012 are projections). Use the data and a graphing calculator to (a) draw a scatterplot

6

108

and decide on an appropriate form of regression, then (b) find the regression equation

and use it to estimate the amount spent by consumers in 2003, (c) the projected

7

128

amount that will be spent in 2014 if this rate of growth continues, and (d) the year

10

267

that \$591 billion is the projected amount of retail spending over the Internet.

11

301

28. The drag force on a compact car driving along the highway on a windless day,

12

335

depends on a constant 2k and the velocity of the car, where k is determined using the

density of the air, the cross-sectional area of the car, and the drag coefficient of the

Velocity

vehicle. The data shown in the table gives the magnitude of the drag force Fd, at given

Fd

(mph)

velocity v. Use the data and a graphing calculator to (a) draw a scatterplot and decide

10

32

on an appropriate form of regression, then (b) find a regression equation and use it to

estimate the magnitude of the drag force for this car at 60 mph. Finally, (c) estimate the

30

306

speed of the car if the drag force has a magnitude of 2329 units.

50

860

29. The graph and accompanying table show the number N of active Starbucks outlets for

70

1694

selected years t from 1990 to 2008. Use the graph and table to (a) find the average rate of

change for the years 1994 to 1996 (the interval [4, 6]). (b) Verify that the rate of growth between the years 2000

and 2002 (the interval [10, 12]) was about 4 times greater than from 1994 to 1996. (c) Show that the average rate

of change for the years 2002 to 2004 was very close to the rate of change for the years 2006 to 2008.

200

Outlets (N)

160

120

Year t

(1990 → 0)

Outlets N

(100s)

Year t

(1990 → 0)

Outlets N

(100s)

0

0.84

10

35.01

2

1.65

12

58.86

80

4

4.25

14

85.69

6

10.15

16

124.40

8

18.86

18

150.79

40

0

2

4

6

8

10

12

14

16

18

20

Year (t)

30. According to Torricelli’s law for tank draining, the volume (in ft3) of a full 5 ft ϫ 2 ft ϫ 2 ft bathtub t sec after

the plug is pulled can be modeled by the function V1t2 ϭ 1Ϫ0.2t ϩ 1202 2. (a) What is the volume of the bathtub

at t ϭ 0 sec? (b) What is the volume of the bathtub at t ϭ 1 sec? (c) What is the average rate of change from t ϭ 0

to t ϭ 1? (d) What is the average rate of change from t ϭ 20 to t ϭ 21? (e) When is the bathtub empty?

SECTION 3.5

The Algebra of Functions

KEY CONCEPTS

• The notation used to represent the basic operations on two functions is

• 1 f ϩ g21x2 ϭ f 1x2 ϩ g1x2

• 1 f Ϫ g2 1x2 ϭ f 1x2 Ϫ g1x2

f 1x2

f

; g1x2 0

• 1 f # g21x2 ϭ f 1x2 # g1x2

• a g b1x2 ϭ

g1x2

• The result of these operations is a new function h(x). The domain of h is the intersection of domains for f and g,

f

excluding values that make g1x2 ϭ 0 for h1x2 ϭ a b1x2 .

g

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