Stewart J. College Algebra: Concepts and Contexts

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494 CHAPTER 6

■

Power, Polynomial, and Rational Functions

y=x∞

y=x¢

y=x£

y=≈

y=x

y y y y

x x x x x

figure 1 Graphs of f 1x 2= x

for p = 1, 2, 3, 4, and 5

In this section we discuss the case where the power p is positive, and in Section

6.5 we will deal with negative powers.

We are already familiar with the cases in which the power is or . If

and , the power function is just , whose graph is the line .

If and , we have the squaring function , whose graph is the

parabola . Figure 1 shows the graphs of the power functions with for the

first several positive integer powers.

C = 1y = x

f 1x2= x

p = 2C = 1

y = xf 1x2= xp = 1C = 1

p = 2p = 1

Notice from Figure 1 that the general shape of the graph of the power function

depends on whether p is odd or even. If p is an even positive integer, then

the graph of is similar to the parabola . If p is an odd positive inte-

ger, then the graph of is similar to that of . Observe from Figure 2,

however, that as p increases, the graph of becomes flatter near 0 and steeper

when and when .x 6-1x 7 1

y = x

f 1x2= x

y = x

f 1x2= x

(1, 1)

y=x∞

y=x£

y=x§

y=x™

y=x¢

(_1, _1)

(_1, 1)

figure 2 Families of power functions

We see from Figures 1 and 2 that for , power functions with larger expo-

nents grow faster than power functions with smaller exponents. In the following ex-

ample we compare growth rates of a power function and an exponential function.

x 7 1

example

Comparing Power and Exponential Functions

Compare the rates of growth of the exponential function and the power

function by drawing the graphs of both functions in the following view-

ing rectangles.

(a)

(b)

30, 64 by 30, 254

30, 34 by 30, 84

g1x 2= x

f 1x2= 2

SECTION 6.2

■

Power Functions: Positive Powers 495

Solution

(a) Figure 3(a) shows that the graph of catches up with, and becomes

higher than, the graph of at .

(b) The larger viewing rectangle in Figure 3(b) shows that the graph of

overtakes that of when .

is much larger than .g1x 2= x

f 1x2= 2

x = 4g1x2= x

f 1x2= 2

x = 2f 1x 2= 2

g1x 2= x

■ NOW TRY EXERCISE 17 ■

(a)

1000

(c)

˝=≈

Ï=2

(b)

˝=≈

Ï=2

˝=≈

Ï=2

figure 3

example

Transformations of Power Functions

Sketch graphs of the following functions.

(a) (b) (c)

Solution

We use the graphs in Figure 2 and transform them using the methods of Section 5.1.

(a) The graph of is the reflection of the graph of in the x-axis,

as shown in Figure 4(a).

(b) The graph of is the graph of shifted to the right 2

units, as shown in Figure 4(b).

stretching the graph vertically and reflecting it in the x-axis (see the dashed

blue graph in Figure 4(c)). Finally, the graph of is obtained

by shifting upward 4 units (see the red graph in Figure 4(c)).

R1x 2=-2x

+ 4

y =-2x

y = x

Q1x 2= 1x - 22

y = x

P1x 2=-x

R1x 2=-2x

+ 4Q1x 2= 1x - 2 2

P1x 2=-x

Q(x)=(x-2)¢

P(x)=_x£

R(x)=_2x∞+4

1_2

(

)(

)

figure 4

■ NOW TRY EXERCISES 7 AND 9 ■

496 CHAPTER 6

■

Power, Polynomial, and Rational Functions

■ Direct Proportionality

We studied the concept of direct proportionality in Section 2.4. In general, two quan-

tities are directly proportional if one is a constant multiple of the other.

Direct Proportionality

If the quantities Q and R are related by the equation

for some nonzero constant k, we say that Q is directly proportional to R.

The constant k is called the constant of proportionality.

Q = kR

It often happens that a quantity is proportional to a power function.

example

Finding the Proportionality Constant

Suppose that L is proportional to the fifth power of w.

(a) Express the proportionality as an equation.

(b) If it is known that when , find the proportionality constant.

Solution

(a) We are given that L is proportional to , so we can write

where k is a constant and .

(b) To find k, we use the given information that when .

L is proportional to

Replace L by 80 and w by 2

Divide by , and switch sides

Calculate

The proportionality constant is , so .

■ NOW TRY EXERCISE 23 ■

L = 2.5w

k = 2.5

k =

80 = k12

L = hw

w = 2L = 80

k ⫽ 0

L = kw

w = 2L = 80

example

The Effect of Doubling the Radius of a Ball

The volume of a sphere with radius r is given by the power function

V1r 2=

SECTION 6.2

■

Power Functions: Positive Powers 497

(a) Describe the relationship between the volume and radius as a proportionality.

(b) What happens to the volume of a ball when you double the radius?

Solution

(a) From the function we see that the volume is directly propor-

tional to the cube of the radius r. The constant of proportionality is .

(b) If you double the radius, it becomes 2r, and the corresponding volume is

Replace r by 2r

Rules of Exponents

Group factors

Replace by (r)

So if you double the radius, the volume (2r) of the ball is 8 times as large as

the volume (r) of the original ball.

■ NOW TRY EXERCISE 35 ■

= 8

V1r 2

= 81

V12r 2=

p12r 2

VV1r2=

Rules of Exponents are

reviewed in Algebra Toolkits

A.3 and A.4, pages T14 and T20.

■ Fractional Positive Powers

If the power p in a power function is a fraction of the form, , where m and

n are positive integers, then

In particular, if , it becomes

and the power function is a root function.

For the root function is the square root function , whose

domain is and whose graph is shown in Figure 5(a). (This graph is actually

the upper half of the parabola .) For other even values of n the graph of

is similar to the graph of the square root function, as in Figure 5(b).f 1x 2= 1

x = y

30, q 2

f 1x2= 1xn = 2

f 1x2= x

1>n

= 1

x˛

p = 1>n

f 1x2= x

= x

m>n

= 1

x˛

p = m>n

Fractional exponents are

reviewed in Algebra Toolkit A.4,

page T20.

figure 5 Graphs of root functions with n evenf 1x 2= 1

(1, 1)

(a) f(x)=œ∑xxœ∑

(1, 1)

(b) f(x)=¢

498 CHAPTER 6

■

Power, Polynomial, and Rational Functions

example

The Inverse Function of a Distance Function

If a stone is dropped from a cliff, Galileo’s Law says that the distance d (in feet) trav-

eled by the stone after t seconds is given by the power function . (So the

distance is proportional to the square of the time.)

(a) Find the inverse function of f.

(b) Interpret the inverse function.

Solution

(a) First we write the function f in equation form, and then we solve for t.

Equation form

Divide by 16

Take square roots

Simplify, and switch sides

So the inverse function is .

(b) The interpretation is that the time required for the stone to fall a distance of

d feet is

This says that the falling time is proportional to the square root of the distance

fallen.

■ NOW TRY EXERCISE 37 ■

t = f˛

-1

1d2=

-1

1d2=

t =

= t

d = 16t

f 1t2= 16t

figure 6 Graphs of root functions with n oddf 1x2= 1

(1, 1) (1, 1)

xœ∑

(b) f(x)=∞xœ∑(a) f(x)=£

Inverse functions are studied in

Section 4.6.

The following example shows how the square root function is used to model the

falling time of a dropped object.

For we have the cube root function , whose domain is the

whole real line. (Recall that every real number has a cube root.) The graphs of

for other odd values of n are similar. (See Figure 6.)f 1x2= 1

f 1x2= 1

xn = 3

SECTION 6.2

■

Power Functions: Positive Powers 499

■ Modeling with Power Functions

We have seen in Example 5 how power functions arise in physics. And in Section 6.4

you will be asked to use data concerning the planets to model the period of revolu-

tion of a planet as a power function of its distance from the sun.

Here we describe how quantities in the biological sciences are modeled by

power functions. In the exercises, power function models are given for the wingspan

of a bird as a function of its weight and for the natural walking speed of an animal

as a function of its leg length. (These models will be derived from data in Section

6.4.) The next example is concerned with the species-area relationship. It’s certainly

plausible that the larger the area of a region, the larger the number of species that in-

habit the region. Many ecologists have modeled the relationship as a direct propor-

tionality involving power functions.

example

The Species-Area Relation in a Bat Cave

The number of species S of bats living in caves in central Mexico has been related to

the surface area A of the interior of the caves by the equation . (In

Example 1 in Section 6.4 this model will be derived from data.)

(a) The cave called Mision Imposible near Puebla, Mexico, has a surface area

of . How many species of bats would you expect to find in that

cave?

(b) If you discover that four species of bats live in a cave, estimate the area of the

cave.

Solution

(a) According to the model , the expected number of species in a

cave with surface area is

So we would expect there to be two species of bats in the Mision Imposible

cave.

(b) For a cave with four species of bats we have

Model

Replace S by 4

Divide by 0.14, and switch sides

Raise each side to the power

Calculator

We predict that a cave with four species of bats would have a surface area of

approximately .

■ NOW TRY EXERCISE 39 ■

188 m

A L 188

1>0.64 A = a

0.14

1>0.64

0.64

0.14

4 = 0.14A

0.64

S = 0.14A

0.64

S = 0.141602

0.64

L 1.92

A = 60 m

S = 0.14A

0.64

A = 60 m

S = 0.14A

0.64

Solving power equations is

reviewed in Algebra Toolkit C.1,

page T47.

Bats in a cave

Grigory Kubatyan/Shutterstock.com 2009

500 CHAPTER 6

■

Power, Polynomial, and Rational Functions

6.2 Exercises

Check your knowledge of solving equations involving a power of the variable by

doing the following problems. You can review these topics in

Algebra Toolkit C.1

on page T47.

1. Simplify each expression.

(a) (b) (c) (d)

2. The given equation involves a power of the variable. Find all real solutions

of the equation.

(a) (b) (c) (d)

3. The given equation involves a power of the variable. Find all real solutions

of the equation.

(a) (b) (c) (d)

4. An equation in two variables is given. Solve the equation for the indicated

variable.

(a) ; y (b) ; S

= 31a

2.12

z9w = z

1>5

= a

x = 2y

2.1S

1.03

= 17.2116y2

3>4

= 16A

4>3

= 81x

1>2

= 5

10.111.1z2

= 20313t 2

= 6x

=-27x

= 25

1>2

3>2

Fundamentals

1. (a) When p is an even positive integer, then the graph of the power function

is similar to the graph of

_______.

(b) When p is an odd positive integer, then the graph of the power function is

similar to the graph of

_______.

2. Portions of the graphs of , , , , and are plotted in the

figures. Match each function with its graph.

y = x

y =

f 1x 2= x

y =

f 1x 2= x

CONCEPTS

3. If the quantities x and y are related by the equation , then we say that y is

_______ proportional to the _______ power of x and the constant of proportionality

_______.

4. If y is directly proportional to the third power of x and if the constant of proportionality

is 4, then x and y are related by the equation .y =

ⵧ

y = 7x

SECTION 6.2

■

Power Functions: Positive Powers 501

Think About It

5. True or false?

(a) For the power function is greater than the power function .

(b) For the power function is greater than the power function

6. Graph the functions , , , and for on the same

coordinate axes. What do you think the graph of would look like on this same

interval? What about ?

7–10

■ Sketch the graph of each function by transforming the graph of an appropriate function

of the form from Figure 1, page 494. Indicate all x- and y-intercepts.

7. (a) (b)

8. (a) (b)

9. (a) (b)

10. (a) (b)

11–16

■ Graph the family of functions in the same viewing rectangle, using the given

values of c. Explain how changing the value of c affects the graph.

11. ; 12. ;

13. ; 14. ;

15. ; 16. ;

17. Compare the rates of growth of the functions and by drawing the

graphs of both functions in the following viewing rectangles.

(a) by (b) by (c) by

18. Compare the rates of growth of the functions and by drawing the

graphs of both functions in the following viewing rectangles.

(a) by (b) by (c) by

19–22

■ Write an equation that expresses the statement.

19. is directly proportional to the third power of x.

20. E is directly proportional to the square of c.

21. is directly proportional to the square root of y.

22. R is directly proportional to the cube root of t.

23–26

■ Express the statement as an equation. Use the given information to find the

constant of proportionality.

23. A is directly proportional to the third power of x. If , then .

24. P is directly proportional to the fourth power of T. If , then .

25. is directly proportional to square root of t. If , then .

26. S is directly proportional to the fifth root of r. If , then .

27–28

■ Use the given information to solve the problem.

27. y is directly proportional to the square of x. If is the constant of proportionality,

then find y when .

28. P is directly proportional to the square root of t. If is the constant of

proportionality, then find P when .t = 64

k = 1.5

x = 4

k = 0.5

S = 12r =

z = 2t = 64z

P = 27T = 3

A = 40x = 2

30, 10

430, 20430, 5000430, 10430, 2043- 4, 44

g1x 2= x

f 1x 2= 3

30, 10

430, 50430, 10

430, 25430, 20430, 54

g1x 2= x

f 1x 2= 2

c =

G1x 2= x

c =

F1x 2= x

c =-1, 0, 1, 2Q1x 2= x

+ cc = 1, 2, 5,

P1x 2= cx

c = 2, 4, 6, 8S1x2= x

c = 1, 3, 5, 7R1x 2= x

S1x 2= 1x - 12

1>5

R1x 2= 1x + 22

1>5

Q1x 2= x

1>4

- 8P1x 2= x

1>4

+ 2

S1x 2=

1x - 12

+ 4R1x 2=-1x + 22

Q1x 2=-x

+ 27P1x2= x

- 8

y = x

101

y = x

100

- 1 … x … 1y = x

y = x

g1x 2= x

f 1x 2= x

0 6 x 6 1

g1x 2= x

f 1x 2= x

x 7 1

SKILLS

x w

0 0

1 1.4

2 2.8

3 5.6

4 11.2

x w

0 0

1 2.1

2 16.8

3 56.7

4 134.4

x w

0 0

1 2

2 16

3 54

4 128

x w

0 51

1 153

2 459

3 1377

4 4131

502 CHAPTER 6

■

Power, Polynomial, and Rational Functions

29. Determine whether the variable w is directly proportional to the third power of x. If so,

write the equation of proportionality.

(a) (b) (c)

31. Power from a Windmill The power P that can be obtained from a windmill is

directly proportional to the cube of the wind speed s.

(a) Write an equation that expresses this proportionality.

(b) Find the constant of proportionality for a windmill that produces 96 watts of power

when the wind is blowing at 20 mi/h.

32. Walking Speed The natural walking speed s of a person or animal is proportional to

the square root of their leg length l.

(a) Write an equation that expresses this proportionality.

(b) A person whose legs are 3 ft long has a natural walking speed of 3.7 ft/s. Find the

constant of proportionality.

33. Skidding in a Curve A 1000-kilogram car is traveling on a curve that forms a

circular arc. The centripetal force F that pushes the car out is directly proportional to the

square of the speed s.

(a) If the car is traveling around the curve at a speed of 15 meters per second, the

centripetal force is 2500 newtons. Find the constant of proportionality, and write an

equation that expresses this proportionality.

(b) The car will skid if the centripetal force is greater than the frictional force holding the

tires to the road. For this road the maximum frictional force is 5880 newtons. If the

car travels around the curve at a speed of 20 meters per second, will it skid?

34. A Jet of Water The power P of a jet of water from a fire hose 3 inches in diameter is

directly proportional to the cube of the water velocity . If the velocity is cut in half, by

what factor will the power decrease?

√

CONTEXTS

x r

0 0

1 5

2 80

3 405

4 1280

x t

0 0

1 2.3

2 4.6

3 6.9

4 9.2

30. Determine whether the variable w is related to the variable x by a linear, exponential, or

power function. In each case, find the function.

(a) (b) (c)

SECTION 6.2

■

Power Functions: Positive Powers 503

35. Radiation Energy The radiation energy E emitted by a heated surface (per unit

surface area) is directly proportional to the fourth power of the absolute temperature T

of the surface.

(a) Express this relationship by writing an equation.

(b) The temperature is 6000 K at the surface of the sun and 300 K at the surface of the

earth. How many times more radiation energy per unit area is produced by the sun

than by the earth?

36. Law of the Pendulum The period of a pendulum (the time elapsed during one

complete swing of the pendulum) is directly proportional to the square root of the

length of the pendulum.

(a) Express this relationship by writing an equation.

(b) To double the period, how would we have to change the length l?

37. Stopping Distance The stopping distance D of a car after the brakes have been

applied is directly proportional to the square of the speed s. The stopping distance of a

car on dry asphalt is given by the function , where s is measured in mi/h.

(a) Find the inverse function of f.

(b) Interpret the inverse function.

applied?

38. Aerodynamic Lift If a certain airplane wing has an area of 500 square feet, then

the lift L of the airplane wing at takeoff is proportional to the square of the speed s

of the plane in miles per hour. The lift of the airplane is given by the function

(a) Find the inverse function of f.

(b) Interpret the inverse function.

f 1s 2= 0.00136s

f 1s 2= 0.033s

Lift

39. Ostrich Flight? The weight W (in pounds) of a bird (that can fly) has been related to

the wingspan L (in inches) of the bird by the equation . (In Exercise 19

of Section 6.4 this model will be derived from data.)

(a) The bald eagle has a wingspan of about 90 inches. Use the model to estimate the

weight of the bald eagle.

(b) An ostrich weighs about 300 pounds. Use the model to estimate what the wingspan

of an ostrich should be in order for it to be able to fly.

explain why ostriches can’t fly.

40. Biodiversity The number N of species of reptiles and amphibians inhabiting

Caribbean islands has been related to the area A (in square meters) of an island by the

equation .

(a) Use the model to estimate the number of species found in an area that is 65 square

meters.

(b) If you discover that 20 species of reptiles and amphibians live on an island,

estimate the area of the island.

N = 3.1046A

0.3080

L = 30.6

0.3952