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C. Graphs of Basic Power Functions

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Section 2.4 Basic Rational Functions and Power Functions; More on the Domain

1

3

5

The functions

y ϭ x2, y ϭ x4, y ϭ x3, y ϭ 1

x, and y ϭ x2 are all power functions,

1

5

4

but only y ϭ x and y ϭ 1 x are also root functions. Initially we will focus on

power functions where p 7 0.

EXAMPLE 5

ᮣ

Comparing the Graphs of Power Functions

Use a graphing calculator

to graph the power functions f 1x2 ϭ x4, g1x2 ϭ x3,

3

1

2

h1x2 ϭ x , p1x2 ϭ x , and q1x2 ϭ x2 in the standard viewing window. Make an

observation in QI regarding the effect

of the 7exponent on each function, then

1

discuss what the graphs of y ϭ x6 and y ϭ x2 would look like.

1

Solution

ᮣ

First we enter the functions in sequence

as Y1 through Y5 on the Y= screen

(Figure 2.61). Using ZOOM 6:ZStandard

produces the graphs shown in

Figure 2.62. Narrowing the window

to focus on QI (Figure 2.63:

x ʦ 3 Ϫ4, 104, y ʦ 3Ϫ4, 10 4 ), we

quickly see that for x Ն 1, larger

values of p cause the graph of y ϭ x p

to increase at a faster rate, and smaller

values at a slower rate. In other words

1

1

(for x Ն 1), since 6 , the graph of

6

4

1

y ϭ x6 would increase slower and appear

1

to be “under” the graph of Y1 ϭ X4.

7

7

Since 7 2, the graph of y ϭ x2 would

2

increase faster and appear to be “more

narrow” than the graph of Y5 ϭ X2

(verify this).

2

Figure 2.61, 2.62

10

Ϫ10

10

Ϫ10

Figure 2.63

10 Y5

Y4

Y3

Y2

Y1

Ϫ4

10

Ϫ4

Now try Exercises 39 through 48

ᮣ

The Domain of a Power Function

In addition to the observations made in Example 5, we can make other important notes,

particularly regarding the domains of power functions. When the exponent on a power

m

7 0 in simplest form, it appears the domain is all real

function is a rational number

n

2

1

numbers if n 2 is odd, as seen in the graphs of g1x2 ϭ x3 , h1x2 ϭ x1 ϭ x1 , and

2

q1x2 ϭ x ϭ x1. If n is an even1 number, the domain

is all nonnegative real numbers as

3

seen in the graphs of f 1x2 ϭ x4 and p1x2 ϭ x2. Further exploration will show that if p is

irrational, as in y ϭ x, the domain is also all nonnegative real numbers and we have

the following:

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CHAPTER 2 More on Functions

The Domain of a Power Function

Given a power function f 1x2 ϭ x p with p 7 0.

m

1. If p ϭ is a rational number in simplest form,

n

a. the domain of f is all real numbers if n is odd: x ʦ 1Ϫq, q 2 ,

b. the domain of f is all nonnegative real numbers if n is even: x ʦ 3 0, q 2 .

2. If p is an irrational number, the domain of f is all nonnegative real numbers:

x ʦ 30, q 2 .

Further confirmation

of statement

1 can be found by recalling the graphs of

1

1

3

y ϭ 1x ϭ x2 and y ϭ 1

x ϭ x3 from Section 2.2 (Figures 2.64 and 2.65).

Figure 2.64

y

Figure 2.65

f(x) ϭ ͙x

5

5

(note n is even)

(9, 3)

(4, 2) (6, 2.4)

(0, 0)

Ϫ1

(note n is odd)

(8, 2)

(1, 1)

(0, 0)

4

8

3

y g(x) ϭ ͙x

x

Ϫ8

Ϫ4

(1, 1)

4

8

x

(Ϫ1, Ϫ1)

(Ϫ8, Ϫ2)

Ϫ5

Ϫ5

Domain: x ʦ 30, q 2

Range: y ʦ 3 0, q 2

EXAMPLE 6

ᮣ

Domain: x ʦ 3Ϫq, q2

Range: y ʦ 3Ϫq, q2

Determining the Domains of Power Functions

State the domain of the following power functions, and identity whether each is

also a root function.

4

1

2

8

a. f 1x2 ϭ x5 b. g1x2 ϭ x10 c. h1x2 ϭ 1

x d. q1x2 ϭ x3 e. r 1x2 ϭ x1 5

Solution

ᮣ

a. Since n is odd, the domain of f is all real numbers; f is not a root function.

b. Since n is even, the domain of g is x ʦ ΄0, q 2 ; g is a root function.

1

c. In exponential form h1x2 ϭ x8. Since n is even, the domain of h is x ʦ ΄0, q 2 ;

h is a root function.

d. Since n is odd, the domain of q is all real numbers; q is not a root function

e. Since p is irrational, the domain of r is x ʦ ΄0, q 2 ; r is not a root function

Now try Exercises 49 through 58

ᮣ

Transformations of Power and Root Functions

As we saw in Section 2.2 (Toolbox Functions and Transformations), the graphs of the

3

root functions y ϭ 1x and y ϭ 1 x can be transformed using shifts, stretches,

reflections, and so on. In Example 8(b) (Section 2.2) we noted the graph of

3

3

h1x2 ϭ 2 1

x Ϫ 2 Ϫ 1 was the graph of y ϭ 1

x shifted 2 units right, stretched by a

factor of 2, and shifted 1 unit down. Graphs of other power functions can be transformed in exactly the same way.

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Section 2.4 Basic Rational Functions and Power Functions; More on the Domain

EXAMPLE 7

ᮣ

Graphing Transformations of Power Functions

Based on our previous observations,

2

3

a. Determine the domain of f 1x2 ϭ x3 and g1x2 ϭ x2 , then verify by graphing

them on a graphing calculator.

2

3

b. Next, discuss what the graphs of F1x2 ϭ 1x Ϫ 22 3 Ϫ 3 and G1x2 ϭ Ϫx2 ϩ 2

will look like, then graph each on a graphing calculator to verify.

Solution

ᮣ

m

a. Both f and g are power functions of the form y ϭ x n . For f, n is odd so its

domain is all real numbers. For g, n is even and the domain is x ʦ 30, q2 .

Their graphs support this conclusion (Figures 2.66 and 2.67).

Figure 2.66

Figure 2.67

10

Ϫ10

10

10

Ϫ10

Ϫ10

10

Ϫ10

b. The graph of F will be the same as the graph of f, but shifted two units right

and three units down, moving the vertex to 12, Ϫ32 . The graph of G will be the

same as the graph of g, but reflected across the x-axis, and shifted 2 units up

(Figures 2.68 and 2.69).

Figure 2.69

Figure 2.68

10

Ϫ10

C. You’ve just seen how

we can graph basic power

functions and state their

domains

10

10

Ϫ10

Ϫ10

10

Ϫ10

Now try Exercises 59 through 62

ᮣ

D. Applications of Rational and Power Functions

These new functions have a variety of interesting and significant applications in the

real world. Examples 8 through 10 provide a small sample, and there are a number of

additional applications in the Exercise Set. In many applications, the coefficients may

be rather large, and the axes should be scaled accordingly.

EXAMPLE 8

ᮣ

Modeling the Cost to Remove Waste

For a large urban-centered county, the cost to remove chemical waste and other

Ϫ18,000

Ϫ 180,

pollutants from a local river is given by the function C1p2 ϭ

p Ϫ 100

where C( p) represents the cost (in thousands of dollars) to remove p percent of

the pollutants.

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a. Find the cost to remove 25%, 50%, and 75% of the pollutants and comment on

the results.

b. Graph the function using an appropriate scale.

c. Use mathematical notation to state what happens as the county attempts to

remove 100% of the pollutants.

Solution

ᮣ

a. We evaluate the function as indicated, finding that C1252 ϭ 60, C1502 ϭ 180,

and C1752 ϭ 540. The cost is escalating rapidly. The change from 25% to 50%

brought a $120,000 increase, but the change from 50% to 75% brought a

$360,000 increase!

C(p)

b. From the context, we need only graph the

x ϭ 100

1200

portion from 0 Յ p 6 100. For the C-intercept

we substitute p ϭ 0 and find C102 ϭ 0, which

900

seems reasonable as 0% would be removed

(75, 540)

600

if $0 were spent. We also note there must be

a vertical asymptote at x ϭ 100, since this

300

x-value causes a denominator of 0. Using

(25, 60)

(50, 180)

p

this information and the points from part (a)

100

50

75

25

produces the graph shown.

c. As the percentage of pollutants removed

y ϭ Ϫ180

approaches 100%, the cost of the cleanup

skyrockets. Using notation: as p S 100 Ϫ , C S q .

Now try Exercises 65 through 70

ᮣ

While not obvious at first, the function C(p) in Example 8 is from the family of

1

reciprocal functions y ϭ . A closer inspection shows it has the form

x

Ϫ18,000

1

Ϫa

Ϫk S

Ϫ 180, showing the graph of y ϭ is shifted right

yϭ

x

xϪh

x Ϫ 100

100 units, reflected across the x-axis, stretched by a factor of 18,000 and shifted 180 units

down (the horizontal asymptote is y ϭ Ϫ180). As sometimes occurs in real-world

applications, portions of the graph were ignored due to the context. To see the full

graph, we reason that the second branch occurs on the opposite side of the vertical and

horizontal asymptotes, and set the window as shown in Figure 2.70. After entering

C(p) as Y1 on the Y= screen and pressing GRAPH , the full graph appears as shown in

Figure 2.71 (for effect, the vertical and horizontal asymptotes were drawn separately

using the 2nd PRGM (DRAW) options).

Figure 2.71

Figure 2.70

2000

200

0

Ϫ2000

Next, we’ll use a root function to model the distance to the horizon from a

given height.

## College algebra graphs models

## B. Translating Written or Verbal Information into a Mathematical Model

## D. Properties of Real Numbers

## A. The Properties of Exponents

## E. The Product of Two Polynomials

## A. Solving Linear Equations Using Properties of Equality

## F. Solving Applications of Basic Geometry

## B. Common Binomial Factors and Factoring by Grouping

## D. Factoring Special Forms and Quadratic Forms

## E. Polynomial Equations and the Zero Product Property

## C. Addition and Subtraction of Rational Expressions

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C. Graphs of Basic Power Functions