copyright © cengage learning. all rights reserved. 2.7 graphs of rational functions
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Copyright © Cengage Learning. All rights reserved.
2.7 Graphs of Rational Functions
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What You Should Learn
• Analyze and sketch graphs of rational functions.
• Sketch graphs of rational functions that have slant asymptotes.
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The Graph of a Rational Functions
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To sketch the graph of a rational function, use the following guidelines.
The Graph of a Rational Functions
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The Graph of a Rational Function
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Example 2 – Sketching the Graph of a Rational Function
Sketch the graph of by hand.
Solution:
y-intercept: because g(0) =
x-intercepts: None because 3 0.
Vertical asymptote: x = 2, zero of denominator
Horizontal asymptote: y = 0, because degree ofN (x) < degree of D (x)
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Example 2 – Solution
Additional points:
By plotting the intercept, asymptotes,and a few additional points, you canobtain the graph shown in Figure 2.46.Confirm this with a graphing utility.
Figure 2.46
cont’d
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Slant Asymptotes
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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.
Slant Asymptotes
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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.
Slant Asymptotes
Figure 2.50
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For instance, by dividing x + 1 into x2 – x, you have
f (x)
= x – 2 +
As x increases or decreases without bound, the remainder term
approaches 0, so the graph of approaches the liney = x – 2, as shown in Figure 2.50
Slant Asymptotes
Slant asymptote ( y = x 2)
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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.
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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 ofN (x) > degree of D (x)
Additional points:
cont’d
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Example 6 – Solution
The graph is shown in Figure 2.51.
Figure 2.51
cont’d