1. Graphs of Rational Functions
2.7
Copyright © Cengage Learning. All rights reserved.

2. 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 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)

7. 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 in Figure Confirm this with a graphing utility.
Figure 2.46

8. 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 To find the equation of a slant asymptote, use long division.
Figure 2.50

11. 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 in Figure 2.50
Slant asymptote ( y = x  2)

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.

14. 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:

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