Section 4.5 Rational Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Objectives For a rational function, find the domain and graph the function, identifying all of the asymptotes. Solve applied problems involving rational functions.
Rational Function A rational function is a function f that is a quotient of two polynomials, that is, where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial. The domain of f consists of all inputs x for which q(x) 0.
Example Consider. Find the domain and graph f. Solution: When the denominator x + 4 = 0, we have x = 4, so the only input that results in a denominator of 0 is 4. Thus the domain is {x|x 4} or ( , 4) ( 4, ). The graph of the function is the graph of y = 1/x translated to the left 4 units.
Vertical Asymptotes For a rational function f(x) = p(x)/q(x), where p(x) and q(x) are polynomials with no common factors other than constants, if a is a zero of the denominator, then x = a is a vertical asymptote for the graph of the function.
Example Determine the vertical asymptotes of the function. Factor to find the zeros of the denominator: x 2 4 = (x + 2)(x 2) Thus the vertical asymptotes are the lines x = 2 and x = 2.
Horizontal Asymptote The line y = b is a horizontal asymptote for the graph of f if either or both of the following are true: When the numerator and the denominator of a rational function have the same degree, the line y = a/b is the horizontal asymptote, where a and b are the leading coefficients of the numerator and the denominator, respectively.
Determining a Horizontal Asymptote When the numerator and the denominator of a rational function have the same degree, the line y = a/b is the horizontal asymptote, where a and b are the leading coefficients of the numerator and the denominator, respectively. When the degree of the numerator of a rational function is less than the degree of the denominator, the x-axis, or y = 0, is the horizontal asymptote. When the degree of the numerator of a rational function is greater than the degree of the denominator, there is no horizontal asymptote.
Example Find the horizontal asymptote: The numerator and denominator have the same degree. The ratio of the leading coefficients is 6/9, so the line y = 2/3 is the horizontal asymptote.
Crossing an Asymptote The graph of a rational function never crosses a vertical asymptote. The graph of a rational function might cross a horizontal asymptote but does not necessarily do so.
Example Graph Vertical asymptotes: x + 3 = 0, so x = 3. The degree of the numerator and denominator is the same. Thus y = 2, is the horizontal asymptote. 1. Draw the asymptotes with dashed lines. 2. Compute and plot some ordered pairs and draw the curve.
Example continued 4/52 00 44 22 8 44 5 55 3.5 77 h(x)h(x)x
Find all the asymptotes of. The line x = 2 is a vertical asymptote. There is no horizontal asymptote because the degree of the numerator is greater than the degree of the denominator. Note that Example
Example (continued) Divide to find an equivalent expression. The line y = 2x 1 is an oblique asymptote.
Occurrence of Lines as Asymptotes of Rational Functions For a rational function f(x) = p(x)/q(x), where p(x) and q(x) have no common factors other than constants: Vertical asymptotes occur at any x-values that make the denominator 0. The x-axis is the horizontal asymptote when the degree of the numerator is less than the degree of the denominator. A horizontal asymptote other than the x-axis occurs when the numerator and the denominator have the same degree.
Occurrence of Lines as Asymptotes of Rational Functions continued An oblique asymptote occurs when the degree of the numerator is 1 greater than the degree of the denominator. There can be only one horizontal asymptote or one oblique asymptote and never both. An asymptote is not part of the graph of the function.
Graphing Rational Functions 1. Find the real zeros of the denominator. Determine the domain of the function and sketch any vertical asymptotes. 2.Find the horizontal or oblique asymptote, if there is one, and sketch it. 3.Find the zeros of the function. The zeros are found by determining the zeros of the numerator. These are the first coordinates of the x-intercepts of the graph. 4.Find f (0). This gives the y-intercept (0, f (0)), of the function. 5.Find other function values to determine the general shape. Then draw the graph.
Example Graph 1. Find the zeros by solving: The zeros are 1/2 and 3, thus the domain excludes these values. The graph has vertical asymptotes at x = 3 and x = 1/2. We sketch these with dashed lines. 2. Because the degree of the numerator is less than the degree of the denominator, the x-axis, y = 0, is the horizontal asymptote.
Example continued 3. To find the zeros of the numerator, we solve x + 3 = 0 and get x = 3. Thus, 3 is the zero of the function, and the pair ( 3, 0) is the x-intercept. 4.We find f(0): Thus (0, 1) is the y-intercept.
Example continued 5. We find other function values to determine the general shape of the graph and then draw the graph. 7/94 11 2 2/3 1 1/2 11 f(x)f(x)x
More Examples Graph the following functions. a) b) c)
Graph a 1.Vertical Asymptote x = 2 2.Horizontal Asymptote y = 1 3.x-intercept (3, 0) 4. y-intercept (0, 3/2)
Graph b 1. Vertical Asymptote x = 3, x = 3 2. Horizontal Asymptote y = 1 3. x-intercepts ( 2.828, 0) 4. y-intercept (0, 8/9)
Graph c 1. Vertical Asymptote x = 1 2. Oblique Asymptote y = x 1 3. x-intercept (0, 0) 4. y-intercept (0, 0)
“Holes in a Graph” Example Graph: Factor the denominator. The domain of the function is: The numerator and denominator have a common factor x – 2. The zeros of the denominator are ‒ 1 and 2, and the zero of the numerator is 2. Since ‒ 1 is the only zero of the denominator that is NOT a zero of the numerator, the graph of the function has x = ‒ 1 as its only vertical asymptote.
“Holes in a Graph” Example-- continued Graph: The expression can be simplified as: The graph of g(x) is the graph of y = 1/(x + 1) with the point where x = 2 is missing. Substitute 2 for x. The “hole” is located at