2.2
Power
Functions
with
Modeling
Copyright © 2011 Pearson, Inc.
What you’ll learn about




Power Functions and Variation
Monomial Functions and Their Graphs
Graphs of Power Functions
Modeling with Power Functions
… and why
Power functions specify the proportional relationships
of geometry, chemistry, and physics.
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 2
Power Function
Any function that can be written in the form
f(x) = k • xa, where k and a are nonzero constants,
is a power function. The constant a is the
power, and k is the constant of variation, or
constant of proportion. We say f(x) varies as
the ath power of x, or f(x) is proportional to the
ath power of x.
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 3
Example Analyzing Power Functions
State the power and constant of variation for the
function f (x) = 4 x , and graph it.
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 4
Example Analyzing Power Functions
State the power and constant of variation for the
function f (x) = 4 x , and graph it.
f (x) = 4 x = x1/4 = 1× x1/4
so the power is 1/4
y
and the constant
of variation is 1.
x
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 5
Monomial Function
Any function that can be written as
f(x) = k or f(x) = k·xn,
where k is a constant and n is a positive integer,
is a monomial function.
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 6
Example Graphing Monomial
Functions
Describe how to obtain the graph of the function f ( x) = 3x from the graph
of g ( x) = x with the same power n.
3
n
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 7
Example Graphing Monomial
Functions
Describe how to obtain the graph of the function f ( x) = 3x from the graph
of g ( x) = x with the same power n.
3
n
We obtain the graph of f ( x) = 3x by vertically stretching the graph of
3
g ( x) = x by a factor of 3. Both are odd functions.
y
3
x
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 8
Graphs of Power Functions
For any power function f(x) = k·xa, one of the
following three things happens when x < 0.
 f is undefined for x < 0.
 f is an even function.
 f is an odd function.
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 9
Graphs of Power Functions
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 10
Example Graphing Power Functions
Describe the portion of the curve that lies in
Quadrant I or IV. Determine whether f is even,
odd, or undefined for x < 0. Graph the function.
()
f ( x ) = -2x
a. f x = 3x
b.
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-2
13
Slide 2.2 - 11
Example Graphing Power Functions
Describe the portion of the curve that lies in
Quadrant I or IV. Determine whether f is even,
odd, or undefined for x < 0. Graph the function.
()
a. f x = 3x
-2
Passes through (1, 3)
Asymptotic to both axes
Even function, symmetric
about the y-axis
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 12
Example Graphing Power Functions
Describe the portion of the curve that lies in
Quadrant I or IV. Determine whether f is even,
odd, or undefined for x < 0. Graph the function.
()
b. f x = -2x
13
Passes through (0, 0) & (1,–2)
Q IV: decreasing & concave
up
Odd function, symmetric
about the origin
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 13
Quick Review
Write the following expressions using only
positive integer powers.
1. x
5/3
2. r -3
3. m
1.5
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 14
Quick Review
Write the following expressions in the form k × x
a
using a single rational number for the power of a.
4. 16x 3
5.
3
x
27
Copyright © 2011 Pearson, Inc.
Slide 2.2 - 15
Quick Review Solutions
Write the following expressions using only
positive integer powers.
1. x
5/3
2. r -3
3
x
5
1
r3
3. m
1.5
Copyright © 2011 Pearson, Inc.
m
3
Slide 2.2 - 16
Quick Review Solutions
Write the following expressions in the form k × x
a
using a single rational number for the power of a.
4. 16x 3
5.
3
x
27
Copyright © 2011 Pearson, Inc.
4x
1
x
3
3
2
1
3
Slide 2.2 - 17