Polynomial and Rational Functions 2 Polynomial and Rational Functions Copyright © Cengage Learning. All rights reserved.

Graphs of Rational Functions 2.7 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.

The Graph of a Rational Functions

The Graph of a Rational Functions To sketch the graph of a rational function, use the following guidelines.

The Graph of a Rational Function

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 of N (x) < degree of D (x)

Example 2 – Solution Additional points: 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.

Slant Asymptotes

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.

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

Slant Asymptotes 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 line y = x – 2, as shown on the previous slide. Slant asymptote ( y = x  2)

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.

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.

Example 6 – Solution y-intercept: (0, 2), because f (0) = 2 cont’d 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) Additional points:

Example 6 – Solution The graph is shown in Figure 2.51. cont’d