1. Slide 3.5- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational Functions Learn the definition of a rational function. Learn to find vertical asymptotes (if any). Learn to find horizontal asymptotes (if any). Learn to graph rational functions. Learn to graph rational functions with oblique asymptotes. Learn to graph a revenue curve. SECTION 3.5 1 2 3 4 5 6

3. Slide 3.5- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 3.5- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 3.5- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 3.5- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 3.5- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley VERTICAL ASYMPTOTES The line with equation x = a is called a vertical asymptote of the graph of a function f if

8. Slide 3.5- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley VERTICAL ASYMPTOTES

9. Slide 3.5- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley VERTICAL ASYMPTOTES

10. Slide 3.5- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 3.5- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 3.5- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 x = –3 and x = 3. The lines with equations x = 3 and x = –3 are the two vertical asymptotes of f (x). c.The denominator x 2 + 1 has no real zeros. Hence, the graph of the rational function h (x) has no vertical asymptotes.

13. Slide 3.5- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 + 2, with a gap (hole) corresponding to x = 2. Solution

14. Slide 3.5- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Rational Function Whose Graph Has a Hole Solution continued

15. Slide 3.5- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 3.5- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Rational Function Whose Graph Has a Hole Solution continued

17. Slide 3.5- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley HORIZONTAL ASYMPTOTES The line with equation y = k is called a horizontal asymptote of the graph of a function f if

18. Slide 3.5- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 3.5- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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, where a n and b m are the leading coefficients of N(x) and D(x), respectively. RULES FOR LOCATING HORIZONTAL ASYMPTOTES

20. Slide 3.5- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Finding the Horizontal Asymptote Find the horizontal asymptotes (if any) of the graph of each rational function. Solution a.Numerator and denominator have degree 1. is the horizontal asymptote.

21. Slide 3.5- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 3.5- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. This step gives 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. Slide 3.5- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 the sign of f (x). Use the sign graphs and test numbers associated with the zeros of N(x) and D(x), to determine where the graph of f is above the x-axis and where it is below the x-axis. 6.Sketch the graph. Plot the points and asymptotes found in steps 1-5 and symmetry to sketch the graph of f.

24. Slide 3.5- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Graphing a Rational Function Sketch the graph of Step 2Find the vertical asymptotes (if any). y-intercept is 0x-intercept is 0 Solution Step 1Find the intercepts. vertical asymptotes are x = 2 and x = –2.

25. Slide 3.5- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Graphing a Rational Function degree of denominator > degree of numerator y = 0 (the x-axis) is the horizontal asymptote Solution continued Step 3Find the horizontal asymptotes (if any). Step 4Test for symmetry. Symmetric with respect to the origin

26. Slide 3.5- 26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Graphing a Rational Function The three zeros 0, –2 and 2 of the numerator and denominator divide the x-axis into four intervals Solution continued Step 5Find the sign of f in the intervals determined by the zeros of the numerator and denominator.

27. Slide 3.5- 27 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Graphing a Rational Function Solution continued

28. Slide 3.5- 28 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Graphing a Rational Function Solution continued Step 6Sketch the graph.

29. Slide 3.5- 29 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 2 + 2 > 0, no x-intercepts vertical asymptotes are x = –2 and x = 1

30. Slide 3.5- 30 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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:

31. Slide 3.5- 31 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Graphing a Rational Function Solution continued

32. Slide 3.5- 32 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Graphing a Rational Function Solution continued Step 6Sketch the graph.

33. Slide 3.5- 33 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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, we have 0. x-intercept and y-intercept are 0.

34. Slide 3.5- 34 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Graphing a Rational Function y = 1 is the horizontal asymptote Solution continued Step 3 degree of den = degree of num Step 4Symmetry. Symmetric with respect to the y- axis Step 5The graph is always above the x-axis, except at x = 0.

35. Slide 3.5- 35 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Graphing a Rational Function Solution continued Step 6Sketch the graph.

36. Slide 3.5- 36 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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).

37. Slide 3.5- 37 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 is –4. Solution Step 1Solve x 2 – 4 = 0, x-intercepts: –2, 2 Vertical asymptote is x = –1.

38. Slide 3.5- 38 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Graphing a Rational Function y = x – 1 is an oblique asymptote. Solution continued Step 3 degree of num > degree of den Step 4Symmetry. None Step 5Sign of f in the intervals determined by the zeros of the numerator and denominator: –2, 2, and –1.

39. Slide 3.5- 39 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Graphing a Rational Function Solution continued

40. Slide 3.5- 40 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Graphing a Rational Function Solution continued Step 6Sketch the graph.

41. Slide 3.5- 41 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

42. Slide 3.5- 42 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

43. Slide 3.5- 43 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Graphing a Revenue Curve Solution continued Here is the graph of y = R(x) for 0 ≤ x ≤ 100.

44. Slide 3.5- 44 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Graphing a Revenue Curve Solution continued c. From the calculator graph of by using the ZOOM and TRACE features, you can see that the tax rate of about 23% produces he maximum tax revenue of about 53.67 billion dollars for the government.