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Copyright © 2007 Pearson Education, Inc.
Slide 4-1
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
4.2 More on Graphs of Rational Functions
Vertical and Horizontal Asymptotes
p( x)
For the rational function f ( x) 
, written in lowest
q( x)
terms, if f ( x)   as x  a, then the line x  a is a
vertical asymptote ; if f ( x)  a as x  , then the
line y  a is a horizontal asymptote .
Copyright © 2007 Pearson Education, Inc.
Slide 4-3
4.2 Finding Asymptotes: Example 1
Example 1 Find the asymptotes of the graph of
x 1
f ( x) 
.
2 x  5x  3
2
Solution
Vertical asymptotes: set denominator
equal to 0 and solve.
2 x  5x  3  0
2
1
(2 x  1)( x  3)  0  x  or x  3
2
The equations of the vertical asymptotes are
x  1 and x  3.
2
Copyright © 2007 Pearson Education, Inc.
Slide 4-4
4.2 Finding Asymptotes: Example 1
Horizontal asymptote: divide each term by the
variable factor of greatest degree, in this case x2.
x 1
1 1

 2
2
2
f ( x)  2x x
 x x
2 x 5x 3 2  5  3
 2 2
2
x x2
x
x
x

As x gets larger and larger,
1 1 5 3
, 2 , , 2 all approach 0.
x x x x
00
0
 0
200 2
Therefore, the line y = 0 is the horizontal asymptote.
Copyright © 2007 Pearson Education, Inc.
Slide 4-5
4.2 Finding Asymptotes: Example 2
Example 2
Find the asymptotes of the graph of
2x  1
f ( x) 
.
x3
Solution
Vertical asymptote: solve the equation x – 3 = 0.
The line x  3 is the vertical asymptote.
Horizontal asymptote: divide each term by x.
2x 1
1

2
x  20  2  2
f ( x)  x x 
x 3
3
1 0 1

1
x x
x
as x  . Therefore, the horizontal asymptote is the line y  2.
Copyright © 2007 Pearson Education, Inc.
Slide 4-6
4.2 Finding Asymptotes: Example 3
Example 3
Find the asymptotes of the graph of
x 1
f ( x) 
.
x2
2
Solution
Vertical asymptote: x  2  0
Horizontal asymptote:

x2
x2 1
1

1

2
2
2
x
x
x
f ( x) 

x 2
1 2

 2
2
2
x x
x x
1
is undefined .
0
Therefore, there is no horizontal asymptote . This occurs when the
degree of the numerator is greater than the degree in the denominato r.
does not approach any real number as x   since
Copyright © 2007 Pearson Education, Inc.
Slide 4-7
4.2 Finding Asymptotes: Example 3
Rewrite f using synthetic division as follows:
x 1
5
f ( x) 
 x2
x2
x2
2
5
For very large values of x ,
is close to 0, and
x2
the graph approaches the line y = x +2. This line is an
oblique asymptote (neither vertical nor horizontal)
for the graph of the function.
Copyright © 2007 Pearson Education, Inc.
Slide 4-8
4.2 Determining Asymptotes
To find asymptotes of a rational function defined by a rational
expression in lowest terms, use the following procedures.
1.
2.
Vertical Asymptotes
Set the denominator equal to 0 and solve for x. If a is a
zero of the denominator but not the numerator, then the
line x = a is a vertical asymptote.
Other Asymptotes Consider three possibilities:
(a) If the numerator has lower degree than the denominator, there is
a horizontal asymptote, y = 0 (x-axis).
(b) If the numerator and denominator have the same degree, and f is
an x n    a0
f ( x) 
, where bn  0,
n
bn x    b0
the horizontal asymptote is the line y 
Copyright © 2007 Pearson Education, Inc.
an
.
bn
Slide 4-9
4.2 Determining Asymptotes
2.
Other Asymptotes (continued)
(c) If the numerator is of degree exactly one greater than
the denominator, there may be an oblique asymptote.
To find it, divide the numerator by the denominator
and disregard any remainder. Set the rest of the
quotient equal to y to get the equation of the
asymptote.
Notes:
i)
ii)
The graph of a rational function may have more than one vertical
asymptote, but can not intersect them.
The graph of a rational function may have only one other nonvertical asymptote, and may intersect it.
Copyright © 2007 Pearson Education, Inc.
Slide 4-10
4.2 Graphing Rational Functions
Let f ( x) 
p( x)
define a rational expression in lowest terms.
q( x)
To sketch its graph, follow these steps.
1.
2.
3.
4.
5.
Find all asymptotes.
Find the x- and y-intercepts.
Determine whether the graph will intersect its nonvertical asymptote by solving f (x) = k where y = k is the
horizontal asymptote, or f (x) = mx + b where
y = mx + b is the equation of the oblique asymptote.
Plot a few selected points, as necessary. Choose an xvalue between the vertical asymptotes and x-intercepts.
Complete the sketch.
Copyright © 2007 Pearson Education, Inc.
Slide 4-11
4.2 Comprehensive Graph Criteria for a
Rational Function
A comprehensive graph of a rational function will
exhibits these features:
1. all intercepts, both x and y;
2. location of all asymptotes: vertical, horizontal,
and/or oblique;
3. the point at which the graph intersects its nonvertical asymptote (if there is such a point);
4. enough of the graph to exhibit the correct end
behavior (i.e. behavior as the graph approaches
its nonvertical asymptote).
Copyright © 2007 Pearson Education, Inc.
Slide 4-12
4.2 Graphing a Rational Function
Example
x 1
Graph f ( x) 
.
2 x  5x  3
2
Solution
Step 1
From Example 1, the vertical asymptotes
have equations x  12 and x  3, and the
horizontal asymptote is the x - axis.
Step 2
x-intercept: solve f (x) = 0
x 1
0
2 x  5x  3
x 1  0
2
x  1
Copyright © 2007 Pearson Education, Inc.
The x - intercept is  1.
Slide 4-13
4.2 Graphing a Rational Function
y-intercept: evaluate f (0)
0 1
1
f ( 0) 

2
2(0)  5(0)  3
3
Step 3
1
The y - intercept is 
3
To determine if the graph intersects the
horizontal asymptote, solve
f ( x)  0.  y - value of horizontal asymptote
Since the horizontal asymptote is the x-axis,
the graph intersects it at the point (–1,0).
Copyright © 2007 Pearson Education, Inc.
Slide 4-14
4.2 Graphing a Rational Function
Step 4
Plot a point in each of the intervals determined
by the x-intercepts and vertical asymptotes,
(,3), (3,1), (1, 1 ), and ( 12 , ) to get an
2
idea of how the graph behaves in each region.
Step 5
Complete the sketch. The graph approaches
its asymptotes as the points become farther
away from the origin.
Copyright © 2007 Pearson Education, Inc.
Slide 4-15
4.2 Graphing a Rational Function That Does
Not Intersect Its Horizontal Asymptote
2x  1
Example Graph f ( x) 
.
x3
Solution Vertical Asymptote: x  3  0
Horizontal Asymptote: y  12

x3

y2
x-intercept: 2 x1  0  2 x  1  0 or x   1
x3
2
2(0)1
y-intercept: f (0)  03   13
Does the graph intersect the horizontal asymptote?
2x  1
2
 2 x  6  2 x  1  No solution.
x3
Copyright © 2007 Pearson Education, Inc.
Slide 4-16
4.2 Graphing a Rational Function That Does
Not Intersect Its Horizontal Asymptote
2x  1
, choose
To complete the graph of f ( x) 
x3
13
points (–4,1) and 6, 3 .
 
Copyright © 2007 Pearson Education, Inc.
Slide 4-17
4.2 Graphing a Rational Function with
an Oblique Asymptote
x 1
Example Graph f ( x) 
.
x2
2
Solution Vertical asymptote: x  2  0
Oblique asymptote:
x 1
2
 x2
5

x2
, therefore the
x2
x2
oblique asymptote is the line y  x  2.
x-intercept: None since x2 + 1 has no real
solutions.
y-intercept: f (0)   12 , so the y - intercept is  12 .
Copyright © 2007 Pearson Education, Inc.
Slide 4-18
4.2 Graphing a Rational Function with
an Oblique Asymptote
Does the graph intersect the oblique asymptote?
x 2 1
 x2
x2
x  1  x  4  No solution .
2
2
To complete the graph, choose the points
2.
4, 17
and

1
,

2
3

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 

Slide 4-19
4.2 Graphing a Rational Function with a
Hole
x 4
.
Example Graph f ( x) 
x2
2
Solution Notice the domain of the function cannot include 2.
Rewrite f in lowest terms by factoring the numerator.
x 2  4 ( x  2)( x  2)
f ( x) 

 x  2 ( x  2)
x2
x2
The graph of f is the graph of
the line y = x + 2 with the
exception of the point with
x-value 2.
Copyright © 2007 Pearson Education, Inc.
Slide 4-20