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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