Sullivan PreCalculus Section 3 Sullivan PreCalculus Section 3.4 Rational Functions II: Analyzing Graphs Objectives Analyze the Graph of a Rational Function Solve Applied Problems Involving Rational Functions

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. If - R(x) = R(-x), there is symmetry with respect to the origin. 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 whole 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 Test at x = -2 Test at x = 1 R(-4) = 6 R(-2) = -2 R(1) = 1 Above x-axis Below x-axis Above x-axis Point: (-4, 6) Point: (-2, -2) Point: (1, 1) g.) Finally, graph the rational function R(x)

x = - 3 (-4, 6) (1, 1) (3, 4/3) y = 2 (-2, -2) (-1, 0) (0, 2/3)

Example: The concentration C of a certain drug in a patients bloodstream t minutes after injection is given by: a.) Find the horizontal asymptote of C(t) Since the degree of the denomination is larger than the degree of the numerator, the horizontal asymptote of the graph of C is y = 0.

b.) What happens to the concentration of the drug as t (time) increases? The horizontal asymptote at y = 0 suggests that the concentration of the drug will approach zero as time increases. c.) Use a graphing utility to graph C(t). According the the graph, when is the concentration of the drug at a maximum? The concentration will be at a maximum five minutes after injection.