1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Rational Functions SECTION Define a rational function. Find vertical asymptotes (if any). Find horizontal asymptotes (if any). Graph rational functions. Graph rational functions with oblique asymptotes. Graph a revenue curve

3 © 2010 Pearson Education, Inc. All rights reserved RATIONAL FUNCTION A function f that can be expressed in the form where the numerator N(x) and the denominator D(x) are polynomials and D(x) is not the zero polynomial, is called a rational function. The domain of f consists of all real numbers for which D(x) ≠ 0.

4 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding the Domain of a Rational Function Find the domain of each rational function. Solution a.The domain of f (x) is the set of all real numbers for which x – 1 ≠ 0; that is, x ≠ 1. In interval notation:

5 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding the Domain of a Rational Function Solution continued b.Find the values of x for which the denominator x 2 – 6x + 8 = 0, then exclude those values from the domain. In interval notation: The domain of g (x) is the set of all real numbers such that x ≠ 2 and x ≠ 4.

6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding the Domain of a Rational Function Solution continued c.The domain of h(x) is the set of all real numbers for which x – 2 ≠ 0; that is, x ≠ 2. In interval notation: The domain of g (x) is the set of all real numbers such that x ≠ 2.

7 © 2010 Pearson Education, Inc. All rights reserved VERTICAL ASYMPTOTES The line with equation x = a is called a vertical asymptote of the graph of a function f if

8 © 2010 Pearson Education, Inc. All rights reserved VERTICAL ASYMPTOTES

9 © 2010 Pearson Education, Inc. All rights reserved VERTICAL ASYMPTOTES

10 © 2010 Pearson Education, Inc. All rights reserved LOCATING VERTICAL ASYMPTOTES OF RATIONAL FUNCTIONS If where the N(x) and D(x) do not have a common factor and a is a real zero of D(x), then the line with equation x = a is a vertical asymptote of the graph of f. is a rational function,

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding Vertical Asymptotes Find all vertical asymptotes of the graph of each rational function. Solution a.No common factors, zero of the denominator is x = 1. The line with equation x = 1 is a vertical asymptote of f (x).

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding Vertical Asymptotes Solution continued b.No common factors. Factoring x 2 – 9 = (x + 3)(x – 3), we see the zeros of the denominator are –3 and 3. The lines with equations x = – 3 and x = 3 are the two vertical asymptotes of g (x). c.The denominator x has no real zeros. Hence, the graph of the rational function h (x) has no vertical asymptotes.

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Rational Function Whose Graph Has a Hole Find all vertical asymptotes of the graph of each rational function. The graph is the line with equation y = x + 3, with a gap (hole) corresponding to x = 3. Solution

14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Rational Function Whose Graph Has a Hole Solution continued The graph has no vertical asymptote.

15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Rational Function Whose Graph Has a Hole Solution continued The graph has a hole at x = –2. However, the graph of g(x) also has a vertical asymptote at x = 2.

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Rational Function Whose Graph Has a Hole Solution continued

17 © 2010 Pearson Education, Inc. All rights reserved HORIZONTAL ASYMPTOTES The line with equation y = k is called a horizontal asymptote of the graph of a function f if

18 © 2010 Pearson Education, Inc. All rights reserved RULES FOR LOCATING HORIZONTAL ASYMPTOTES Let f be a rational function given by where N(x) and D(x) have no common factors. Then whether the graph of f has one horizontal asymptote or no horizontal asymptote is found by comparing the degree of the numerator, n, with that of the denominator, m:

19 © 2010 Pearson Education, Inc. All rights reserved 1.If n < m, then the x-axis (y = 0) is the horizontal asymptote. 3.If n > m, then the graph of f has no horizontal asymptote. 2.If n = m, then the line with equation is the horizontal asymptote. RULES FOR LOCATING HORIZONTAL ASYMPTOTES

20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Horizontal Asymptote Find the horizontal asymptote (if any) of the graph of each rational function. Solution a.Numerator and denominator have degree 1. is the horizontal asymptote.

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Horizontal Asymptote Solution continued degree of denominator > degree of numerator y = 0 (the x-axis) is the horizontal asymptote. degree of numerator > degree of denominator The graph has no horizontal asymptote.

22 © 2010 Pearson Education, Inc. All rights reserved PROCEDURE FOR GRAPHING A RATIONAL FUNCTION 1.Find the intercepts. The x-intercepts are found by solving the equation N(x) = 0. The y-intercept is f (0). 2.Find the vertical asymptotes (if any). Solve D(x) = 0 to find the vertical asymptotes of the graph. Sketch the vertical asymptotes. 3.Find the horizontal asymptotes (if any). Use the rules found in an earlier slide.

23 © 2010 Pearson Education, Inc. All rights reserved 4.Test for symmetry. If f (–x) = f (x), then f is symmetric with respect to the y-axis. If f (–x) = – f (x), then f is symmetric with respect to the origin. 5.Find additional graph points. Find x- values for additional graph points by using the zeros of N(x) and D(x). Choose one x- value less than the smallest zero, one x- value between each two consecutive zeros, and one x-value larger than the largest zero. Calculate the corresponding y-values to determine where the graph of f is above or below the x-axis.

24 © 2010 Pearson Education, Inc. All rights reserved 6.Sketch the graph. Plot the points and asymptotes found in steps 1-5 and symmetry to sketch the graph of f.

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing a Rational Function Sketch the graph of Solution Step 1Find the intercepts. The x-intercepts are  1 and 1, so the graph passes through (  1, 0) and (1, 0).

26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing a Rational Function Step 2Find the vertical asymptotes (if any). vertical asymptotes are x = 3 and x = –3. Solution continued The y-intercept is, so the graph of f passes through the point (0, ).

27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing a Rational Function Since n = m = 2 the horizontal asymptote is Solution continued Step 3Find the horizontal asymptotes (if any).

28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing a Rational Function Solution continued Step 4Test for symmetry. The graph of f is symmetric in the y-axis.

29 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing a Rational Function The zeros of N(x) and D(x) are –3, –1, 1, and 3. They divide the x-axis into the five intervals shown on the next slide. Solution continued Step 5Find additional graph points.

30 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing a Rational Function Solution continued

31 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Graphing a Rational Function Solution continued Step 6Sketch the graph.

32 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Graphing a Rational Function Sketch the graph of Step 2Solve (x + 2)(x – 1) = 0; x = –2, x = 1 ; y-intercept is –1. Solution Step 1Since x > 0, no x-intercepts. Vertical asymptotes are x = –2 and x = 1.

33 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Graphing a Rational Function y = 1 is the horizontal asymptote Solution continued Step 3 degree of den = degree of num Step 4Symmetry. None Step 5The zeros of the denominator –2 and 1 yield the following figure:

34 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Graphing a Rational Function Solution continued

35 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Graphing a Rational Function Solution continued Step 6Sketch the graph.

36 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Graphing a Rational Function Sketch a graph of Step 2Because x 2 +1 > 0 for all x, the domain is the set of all real numbers. Since there are no zeros for the denominator, there are no vertical asymptotes. Solution Step 1Since f (0) = 0 and setting f (x) = 0 yields 0, x-intercept and y-intercept are 0.

37 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Graphing a Rational Function Solution continued Step 3degree of den = degree of num y = 1 is the horizontal asymptote. Step 4Symmetry. Symmetric with respect to the y-axis Step 5The graph is always above the x-axis, except at x = 0.

38 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Graphing a Rational Function Solution continued Step 6Sketch the graph.

39 © 2010 Pearson Education, Inc. All rights reserved OBLIQUE ASUMPTOTES Suppose is greater than the degree of D(x). Then and the degree of N(x) Thus, as That is, the graph of f approaches the graph of the oblique asymptote defined by Q(x).

40 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Graphing a Rational Function with an Oblique Asymptote Sketch the graph of Step 2Solve x + 1 = 0; x = –1; domain is set of all real numbers except –1. ; y-intercept: –4. Solution Step 1Solve x 2 – 4 = 0; x-intercepts: –2, 2 Vertical asymptote is x = –1.

41 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Graphing a Rational Function y = x – 1 is an oblique asymptote. Solution continued Step 3 degree of numerator > degree of den Step 4Symmetry. None Step 5Sign of f in the intervals determined by the zeros of the numerator and denominator: –2, –1, and 2

42 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Graphing a Rational Function Solution continued

43 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Graphing a Rational Function Solution continued Step 6Sketch the graph.

44 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Graphing a Revenue Curve The revenue curve for an economy of a country is given by a.Find and interpret R(10), R(20), R(30), R(40), R(50), and R(60). b.Sketch the graph of y = R(x) for 0 ≤ x ≤ 100. c.Use a graphing calculator to estimate the tax rate that yields the maximum revenue. where x is the tax rate in percent and R(x) is the tax revenue in billions of dollars.

45 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Graphing a Revenue Curve Solution If income is taxed at a rate of 10%, total revenue for the government will be 45 billion dollars.

46 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Graphing a Revenue Curve Solution continued Here is the graph of y = R(x) for 0 ≤ x ≤ 100.

47 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Graphing a Revenue Curve Solution continued c. From the calculator graph of by using the TRACE feature, you can see that the tax rate of about 23% produces the maximum tax revenue of about $53.7 billion dollars for the government.