6Example 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 denominatorHorizontal asymptote: y = 0, because degree of N (x) < degree of D (x)
7Example 2 – Solution Additional points: cont’dAdditional 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
9Slant AsymptotesConsider 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.
10Slant 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
11Slant 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 termapproaches 0, so the graph of approaches the line y = x – 2, as shown in Figure 2.50Slant asymptote ( y = x 2)
12Example 6 – A Rational Function with a Slant Asymptote Sketch the graph ofSolution: First write f (x) in two different ways. Factoring the numeratorenables you to recognize the x-intercepts.
13Example 6 – A Rational Function with a Slant Asymptote Long divisionenables you to recognize that the line y = x is a slant asymptote of the graph.
14Example 6 – Solution y-intercept: (0, 2), because f (0) = 2 cont’dy-intercept: (0, 2), because f (0) = 2x-intercepts: (–1, 0) and (2, 0)Vertical asymptote: x = 1, zero of denominatorHorizontal asymptote: None, because degree of N (x) > degree of D (x)Additional points:
15Example 6 – Solution The graph is shown in Figure 2.51. cont’d