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SAT Question of the Day. 8.2 Rational Functions and Their Graphs. Objectives: Identify and evaluate rational functions Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph. acid. solution. add x milliliters of distilled water.
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8.2 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph
acid solution add x milliliters of distilled water Example 1 William begins with 75 milliliters of a 15% acid solution. He adds x milliliters of distilled water to the container holding the acid solution. a) Write a function, C, that represents the acid concentration of the solution in terms of x. 15% of 75 = 11.25
Example 1 William begins with 75 milliliters of a 15% acid solution. He adds x milliliters of distilled water to the container holding the acid solution. b) What is the acid concentration of the solution if 35 milliliters of distilled water is added?
Example 2 Find the domain of Find the values of x for which the denominator equals 0. x2 – 9x – 36 = 0 (x – 12)(x + 3) = 0 x = 12 or -3 The domain is all real numbers except 12 and -3.
Vertical Asymptote In a rational function R, if x – a is a factor of the denominator but not a factor of the numerator, x = a is vertical asymptote of the graph of R. Hole In a rational function R, if x – a is a factor of the denominator AND a factor of the numerator, x = a is a hole in the graph of R.
Example 3 Identify all vertical asymptotes of Factor the denominator. Equations for the vertical asymptotes are x = 2 and x = 1.
R(x) = is a rational function; P and Q are polynomials P Q Horizontal Asymptote • If degree of P < degree of Q, thenthe horizontal asymptote of R is y = 0.
R(x) = is a rational function; P and Q are polynomials P a Q b • If degree of P = degree of Q and a and b are the leading coefficients of P and Q, thenthe horizontal asymptote of R is y = . Horizontal Asymptote
R(x) = is a rational function; P and Q are polynomials P Q Horizontal Asymptote • If degree of P > degree of Q, thenthere is no horizontal asymptote
Homework Lesson 8.2 exercises 11-39 odd
8.2.2 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph
Example 1 Let . Identify all vertical asymptotes and all horizontal asymptotes. Equations for the vertical asymptotes are x = -5 and x = 4. Because the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asymptotes.
Example 2 Let . Identify all vertical asymptotes and all horizontal asymptotes. 1 Vertical asymptotes: x = -3 and x = 3 Horizontal asymptotes: numerator and denominator have the same degree leading coefficients y = 2
Holes in Graphs In a rational function R, if x – b is a factor of the numerator and the denominator, there is a hole in the graph of R when x = b (unless x = b is a vertical asymptote).
f(x) = 2x2 + 2x x2 – 1 2x(x + 1) f(x) = (x + 1)(x – 1) Example 3 Identify all asymptotes and holes in the graph of the rational function. factor: hole in the graph:x = –1 vertical asymptote:x = 1 horizontal asymptote:y = 2
Example 4 Use asymptotes to graph the rational function. Write equations for the asymptotes, and graph them as dashed lines. horizontal asymptote:y = 2 vertical asymptote:x = -4 Use a table to help obtain an accurate plot.
Collins Type 2 Explain how to use the asymptotes of the graph of g(x) = x - 5 x - 3 to sketch the graph of the function.
Homework Worksheet “More problems on rational function graphing”