Sullivan Algebra and Trigonometry: Section 4.4 Rational Functions II: Analyzing Graphs Objectives Analyze the Graph of a Rational Function

To analyze the graph of a rational function: a.) Find the Domain of the rational function. b.) Locate the intercepts, if any, of the graph. c.) Test for Symmetry. If R(-x) = R(x), there is symmetry with respect to the y-axis. d.) Write R in lowest terms and find the real zeros of the denominator, which are the vertical asymptotes. e.) Locate the horizontal or oblique asymptotes. f.) Determine where the graph is above the x-axis and where the graph is below the x-axis. g.) Use all found information to graph the function.

Example: Analyze the graph of

a.) x-intercept when x + 1 = 0: ( – 1,0) b.) y-intercept when x = 0: y – intercept: (0, 2/3) c.) Test for Symmetry: No symmetry

d.) Vertical asymptote: x = – 3 Since the function isn’t defined at x = 3, there is a hole at that point. e.) Horizontal asymptote: y = 2 f.) Divide the domain using the zeros and the vertical asymptotes. The intervals to test are:

Test at x = – 4 R( – 4) = 6 Above x-axis Point: ( – 4, 6) Test at x = – 2 R( – 2) = – 2 Below x-axis Point: (-2, -2) Test at x = 1 R(1) = 1 Above x-axis Point: (1, 1) g.) Finally, graph the rational function R(x)

(-4, 6) (-2, -2) (-1, 0)(0, 2/3) (1, 1) (3, 4/3) There is a HOLE at this Point. y = 2 x = - 3