Chapter 4: Rational, Power, and Root Functions

Copyright © 2007 Pearson Education, Inc. Slide 4-2

Chapter 4: Rational, Power, and Root Functions

4.1 Rational Functions and Graphs

4.2 More on Graphs of Rational Functions

4.3 Rational Equations, Inequalities, Applications, and Models

4.4 Functions Defined by Powers and Roots

4.5 Equations, Inequalities, and Applications Involving Root Functions


4.4 Functions Defined by Powers and Roots

• f (x) = xp/q, p/q in lowest terms– if q is odd, the domain is all real numbers– if q is even, the domain is all nonnegative real

numbers

Power and Root Functions

A function f given by f (x) = xb, where b is a constant, is a power function. If , for some integer n 2, then f is a root function given by f (x) = x1/n, or equivalently, f (x) =

nb 1

.n x


4.4 Graphing Power Functions

Example Graph f (x) = xb, b = .3, 1, and 1.7, for

x 0.

Solution The larger values of b cause the graph of

f to increase faster.


4.4 Modeling Wing Size of a Bird

Example Heavier birds have larger wings with more surface area. For some species of birds, this relationship can be modeled by S (x) = .2x2/3, where x is the weight of the bird inkilograms and S is the surface area of the wings in square meters. Approximate S(.5) and interpret the result.

Solution

The wings of a bird that weighs .5 kilogram have a surface area of about .126 square meter.

126.)5(.2.)5(. 3/2

S


4.4 Modeling the Length of a Bird’s Wing

Example The table lists the weight W and the wingspan L for birds of a particular species.

(a) Use power regression to model the data with L = aWb. Graph the data and the equation.

(b) Approximate the wingspan for a bird weighing 3.2 kilograms.

.5 1.5 2.0 2.5 3.0

.77 1.10 1.22 1.31 1.40

W (in kilograms)

L (in meters)



Solution(a) Let x be the weight W and y be the length L.

Enter the data, and then select power regression (PwrReg), as shown in the following figures.



The resulting equation and graph can be seen in the figures below.

(b) If a bird weighs 3.2 kg, this model predicts the wingspan to be

meters. 42.1)2.3(9674. 3326. L


4.4 Graphs of Root Functions: Even Roots


4.4 Graphs of Root Functions: Odd Roots


4.4 Finding Domains of Root Functions

Example Find the domain of each function.

(a) (b)

Solution

(a) 4x + 12 must be greater than or equal to 0 since the root, n = 2, is even.

(b) Since the root, n = 3, is odd, the domain of g is all real numbers.

124)( xxf 3 88)( xxg

30124

x

xThe domain of f is [–3,).


4.4 Transforming Graphs of Root Functions

Example Explain how the graph of can be obtained from the graph of

Solution

124 xy.xy

32)3(4

124

xx

xy

Shift left 3 units and stretch vertically by a factor of 2.

xy


4.4 Transforming Graphs of Root Functions

Example Explain how the graph of can be obtained from the graph of

Solution

3 88 xy.3 xy

3

3

3

12)1(8

88

xx

xy

Shift right 1 unit, stretch vertically by a factor of 2, and reflect across the x-axis.

3 xy


4.4 Graphing Circles Using Root Functions

• The equation of a circle centered at the origin with radius r is found by finding the distance from the origin to a point (x,y) on the circle.

• The circle is not a function, so imagine a semicircle on top and another on the bottom.

222

222

22

)0()0()0()0(

yxryxr

yxr


4.4 Graphing Circles Using Root Functions

• Solve for y:

• Since y2 = –y1, the “bottom” semicircle is a reflection of the “top” semicircle.

22

222

222

xry

xryryx

semicircle bottom

222

semicircle top

221 and xryxry


4.4 Graphing a Circle

Example Use a calculator in function mode to graph the circle

Solution This graph can be obtained by graphing

in the same

window.

.422 yx

212

21 4and4 xyyxy

Technology Note: Graphs may not connect when using a non-decimal window.

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Chapter 4: Rational, Power, and Root Functions