Presentation on theme: "Sullivan PreCalculus Section 3"— Presentation transcript:
1 Sullivan PreCalculus Section 3 Sullivan PreCalculus Section 3.4 Rational Functions II: Analyzing GraphsObjectivesAnalyze the Graph of a Rational FunctionSolve Applied Problems Involving Rational Functions
2 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.
4 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
5 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 = 2f.) Divide the domain using the zeros and the vertical asymptotes. The intervals to test are:
6 Test at x = -4Test at x = -2Test at x = 1R(-4) = 6R(-2) = -2R(1) = 1Above x-axisBelow x-axisAbove x-axisPoint: (-4, 6)Point: (-2, -2)Point: (1, 1)g.) Finally, graph the rational function R(x)
8 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.
9 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.