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
Copyright © 2007 Pearson Education, Inc. Slide 4-3
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
Copyright © 2007 Pearson Education, Inc. Slide 4-4
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.
Copyright © 2007 Pearson Education, Inc. Slide 4-5
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
Copyright © 2007 Pearson Education, Inc. Slide 4-6
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)
Copyright © 2007 Pearson Education, Inc. Slide 4-7
4.4 Modeling the Length of a Bird’s Wing
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.
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4.4 Modeling the Length of a Bird’s Wing
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
Copyright © 2007 Pearson Education, Inc. Slide 4-9
4.4 Graphs of Root Functions: Even Roots
Copyright © 2007 Pearson Education, Inc. Slide 4-10
4.4 Graphs of Root Functions: Odd Roots
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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,).
Copyright © 2007 Pearson Education, Inc. Slide 4-12
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
Copyright © 2007 Pearson Education, Inc. Slide 4-13
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
Copyright © 2007 Pearson Education, Inc. Slide 4-14
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
Copyright © 2007 Pearson Education, Inc. Slide 4-15
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
Copyright © 2007 Pearson Education, Inc. Slide 4-16
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.