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 To sketch the graph of a rational function, use the following guidelines. The Graph of a Rational Functions
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: By plotting the intercept, asymptotes, and a few additional points, you can obtain the graph shown in Figure 2.46. Confirm this with a graphing utility. Figure 2.46 cont’d
8 Slant Asymptotes
9 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
10 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
11 For instance, by dividing x + 1 into x 2 – 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 Asymptotes 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 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: cont’d
15 Example 6 – Solution The graph is shown in Figure 2.51. Figure 2.51 cont’d