Transcript Ch 4 Rational, Power, and Root Functions

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 4-2
4.4 Functions Defined by Powers and
Roots
Power and Root Functions
A function f given by f (x) = xb, where b is a
constant, is a power function. If b  1n , for some
integer n  2, then f is a root function given by
f (x) = x1/n, or equivalently, f (x) = n x .
•
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
Copyright © 2007 Pearson Education, Inc.
Slide 4-3
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 4-4
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 in
kilograms and S is the surface area of the wings in square
meters. Approximate S(.5) and interpret the result.
Solution S (.5)  .2(.5) 2 / 3
 .126
The wings of a bird that weighs .5 kilogram have a surface
area of about .126 square meter.
Copyright © 2007 Pearson Education, Inc.
Slide 4-5
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.
W (in kilograms)
L (in meters)
.5
.77
1.5 2.0 2.5 3.0
1.10 1.22 1.31 1.40
(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.
Copyright © 2007 Pearson Education, Inc.
Slide 4-6
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.
Copyright © 2007 Pearson Education, Inc.
Slide 4-7
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
L  .9674(3.2).3326  1.42 meters.
Copyright © 2007 Pearson Education, Inc.
Slide 4-8
4.4 Graphs of Root Functions: Even Roots
Copyright © 2007 Pearson Education, Inc.
Slide 4-9
4.4 Graphs of Root Functions: Odd Roots
Copyright © 2007 Pearson Education, Inc.
Slide 4-10
4.4 Finding Domains of Root Functions
Example Find the domain of each function.
3
g
(
x
)

 8x  8
f
(
x
)

4
x

12
(a)
(b)
Solution
(a) 4x + 12 must be greater than or equal to 0 since
the root, n = 2, is even.
4 x  12  0
x  3 The domain of f is [–3,).
(b) Since the root, n = 3, is odd, the domain of g is
all real numbers.
Copyright © 2007 Pearson Education, Inc.
Slide 4-11
4.4 Transforming Graphs of Root
Functions
Example Explain how the graph of y  4 x  12
can be obtained from the graph of y  x.
Solution
y  4 x  12
 4( x  3)
 2 x3
Shift y  x left 3
units and stretch
vertically by a factor
of 2.
Copyright © 2007 Pearson Education, Inc.
Slide 4-12
4.4 Transforming Graphs of Root
Functions
Example
Explain how the graph of y  3  8 x  8
can be obtained from the graph of y  3 x.
Solution
y  3  8x  8
 3  8( x  1)
 2 3 x  1
Shift y  3 x right 1
unit, stretch vertically by
a factor of 2, and reflect
across the x-axis.
Copyright © 2007 Pearson Education, Inc.
Slide 4-13
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.
r  ( x  0) 2  ( y  0) 2
r 2  ( x  0) 2  ( y  0) 2
2
2
2
r x y
• The circle is not a function, so imagine a
semicircle on top and another on the bottom.
Copyright © 2007 Pearson Education, Inc.
Slide 4-14
4.4 Graphing Circles Using Root Functions
•
Solve for y:
x2  y 2  r 2
y 2  r 2  x2
y   r 2  x2
2
2
y1  
r

x

 and
top semicircle
•
2
2
y2  
r

x

bottom semicircle
Since y2 = –y1, the “bottom” semicircle is a
reflection of the “top” semicircle.
Copyright © 2007 Pearson Education, Inc.
Slide 4-15
4.4 Graphing a Circle
Example Use a calculator in function mode to
graph the circle x 2  y 2  4.
Solution This graph can be obtained by graphing
y1  4  x 2 and y2   y1   4  x 2 in the same
window.
Technology Note:
Graphs may not
connect when using a
non-decimal window.
Copyright © 2007 Pearson Education, Inc.
Slide 4-16