2
Polynomial and Rational
Functions
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2.7
Graphs of Rational
Functions
Copyright © Cengage Learning. All rights reserved.
What You Should Learn
•
•
Analyze and sketch graphs of rational functions.
Sketch graphs of rational functions that have
slant asymptotes.
3
The Graph of a Rational Functions
4
The Graph of a Rational Functions
To sketch the graph of a rational function, use the following
guidelines.
5
The Graph of a Rational Function
6
Example 2 – Sketching the Graph of a Rational Function
Sketch the graph of
Solution:
y-intercept:
by hand.
because g(0) =
x-intercepts:
None because 3  0.
Vertical asymptote:
x = 2, zero of denominator
Horizontal asymptote:
y = 0, because degree of
N(x) < degree of D(x)
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Example 2 – Solution
cont’d
Additional points:
By plotting the intercept, asymptotes,
and a few additional points, you can
obtain the graph shown.
Confirm this with a graphing utility.
Note: When you are picking values for
your table, pick #s to the left and right
of asymptotes and between them if you have two.
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Slant Asymptotes
9
Slant Asymptotes
Consider a rational function whose denominator is of
degree 1 or greater.
If the degree of the numerator is exactly one more than the
degree of the denominator, then the graph of the function
has a slant (or oblique) asymptote.
10
Slant Asymptotes
For example, the graph of
has a slant asymptote, as
shown in Figure 2.50. To
find the equation of a slant
asymptote, use long division.
Figure 2.50
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Slant Asymptotes
For instance, by dividing x + 1 into x2 – x, you have
f(x)
=x–2+
Slant asymptote
( y = x  2)
As x increases or decreases without bound, the remainder
term
approaches 0, so the graph of approaches the line
y = x – 2, as shown on the previous slide.
12
Example 6 – A Rational Function with a Slant Asymptote
Sketch the graph of
.
Solution:
First write f(x) in two different ways. Factoring the
numerator
enables you to recognize the x-intercepts.
13
Example 6 – A Rational Function with a Slant Asymptote
Long division
enables you to recognize that the line y = x is a slant
asymptote of the graph.
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Example 6 – Solution
y-intercept:
(0, 2), because f(0) = 2
x-intercepts:
(–1, 0) and (2, 0)
Vertical asymptote:
x = 1, zero of denominator
Horizontal asymptote:
None, because degree of
N(x) > degree of D(x)
cont’d
Additional points:
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Example 6 – Solution
cont’d
The graph is shown in Figure 2.51.
Figure 2.51
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