1. Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 3.5 Rational Functions and Their Graphs

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Find the domains of rational functions. Use arrow notation. Identify vertical asymptotes. Identify horizontal asymptotes. Use transformations to graph rational functions. Graph rational functions. Identify slant asymptotes. Solve applied problems involving rational functions. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Arrow Notation We use arrow notation to describe the behavior of some functions.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Definition of a Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Definition of a Vertical Asymptote (continued) The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Finding the Vertical Asymptotes of a Rational Function Find the vertical asymptotes, if any, of the graph of the rational function: There are no common factors in the numerator and the denominator. The zeros of the denominator are –1 and 1. Thus, the lines x = –1 and x = 1 are the vertical asymptotes for the graph of f.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Finding the Vertical Asymptotes of a Rational Function (continued) Find the vertical asymptotes, if any, of the graph of the rational function: The zeros of the denominator are –1 and 1. Thus, the lines x = –1 and x = 1 are the vertical asymptotes for the graph of f.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Finding the Vertical Asymptotes of a Rational Function Find the vertical asymptotes, if any, of the graph of the rational function: We cannot factor the denominator of h(x) over the real numbers. The denominator has no real zeros. Thus, the graph of h has no vertical asymptotes.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Locating Horizontal Asymptotes

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function: The degree of the numerator, 2, is equal to the degree of the denominator, 2. The leading coefficients of the numerator and the denominator, 9 and 3, are used to obtain the equation of the horizontal asymptote. The equation of the horizontal asymptote is or y = 3.

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function: The equation of the horizontal asymptote is y = 3.

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function: The degree of the numerator, 1, is less than the degree of the denominator, 2. Thus, the graph of g has the x-axis as a horizontal asymptote. The equation of the horizontal asymptote is y = 0.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Finding the Horizontal Asymptote of a Rational Function (continued) Find the horizontal asymptote, if any, of the graph of the rational function: The equation of the horizontal asymptote is y = 0.

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Graphing a Rational Function Graph: Step 1 Determine symmetry. Because f(–x) does not equal either f(x) or –f(x), the graph has neither y-axis symmetry nor origin symmetry.

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Graphing a Rational Function (continued) Graph: Step 2 Find the y-intercept. Evaluate f(0). Step 3 Find x-intercepts.

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Graphing a Rational Function (continued) Graph: Step 4 Find the vertical asymptote(s). Step 5 Find the horizontal asymptote. The numerator and denominator of f have the same degree, 1. The horizontal asymptote is

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Graphing a Rational Function (continued) Graph: Step 6 Plot points between and beyond each x-intercept and vertical asymptotes. The x-intercept is (1,0) The y-intercept is vertical asymptote x = 2 We will evaluate f at

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Graphing a Rational Function (continued) Graph: Step 6 (continued) We evaluate f at the selected points.

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Graphing a Rational Function (continued) Graph: Step 7 Graph the function vertical asymptote x = 2 horizontal asymptote y = 3

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is one more than the degree of the denominator. In general, if p and q have no common factors, and the degree of p is one greater than the degree of q, find the slant asymptotes by dividing q(x) into p(x).

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Example: Finding the Slant Asymptotes of a Rational Function Find the slant asymptote of The degree of the numerator, 2, is exactly one more than the degree of the denominator, 1. To find the equation of the slant asymptote, divide the numerator by the denominator.

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Finding the Slant Asymptotes of a Rational Function Find the slant asymptote of The equation of the slant asymptote is y = 2x – 1. The graph of f(x) is shown. vertical asymptote x = 2 slant asymptote y = 2x – 1

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23

24. Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 3.6 Polynomial and Rational Inequalities

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Solve polynomial inequalities. Solve rational inequalities. Solve problems modeled by polynomial or rational inequalities. Objectives:

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Definition of a Polynomial Inequality A polynomial inequality is any inequality that can be put into one of the forms where f is a polynomial function.

27. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Procedure for Solving Polynomial Inequalities 1. Express the inequality in the form f(x) 0, where f is a polynomial function. 2. Solve the equation f(x) = 0. The real solutions are boundary points. 3. Locate these boundary points on a number line, thereby dividing the number line into intervals.

28. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28 Procedure for Solving Polynomial Inequalities (continued) 4. Choose one representative number, called a test value, within each interval and evaluate f at that number. a. If the value of f is positive, then f(x) > 0 for all numbers, x, in the interval. b. If the value of f is negative, then f(x) < 0 for all numbers, x, in the interval. 5. Write the solution set, selecting the interval or intervals that satisfy the given inequality.

29. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29 Procedure for Solving Polynomial Inequalities (continued) This procedure is valid if is replaced by However, if the inequality involves or include the boundary points [the solutions of f(x) = 0] in the solution set.

30. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 30 Example: Solving a Polynomial Inequality Solve and graph the solution set: Step 1 Express the inequality in the form f(x) > 0 or f(x) < 0 Step 2 Solve the equation f(x) = 0.

31. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 31 Example: Solving a Polynomial Inequality (continued) Solve and graph the solution set: Step 3 Locate the boundary points on a number line and separate the line into intervals. The boundary points divide the line into three intervals

32. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 32 Example: Solving a Polynomial Inequality (continued) Solve and graph the solution set: Step 4 Choose one test value within each interval and evaluate f at that number. IntervalTest Value Substitute intoConclusion –5f(x) > 0 for all x in

33. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 33 Example: Solving a Polynomial Inequality Solve and graph the solution set: Step 4 Choose one test value within each interval and evaluate f at that number. IntervalTest ValueSubstitute intoConclusion 0f(x) < 0 for all x in

34. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 34 Example: Solving a Polynomial Inequality Solve and graph the solution set: Step 4 Choose one test value within each interval and evaluate f at that number. IntervalTest ValueSubstitute intoConclusion 6f(x) > 0 for all x in

35. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 35 Example: Solving a Polynomial Inequality Solve and graph the solution set: Step 5 Write the solution set, selecting the interval or intervals that satisfy the given inequality. Based on Step 4, we see that f(x) > 0 for all x in and Thus, the solution set of the given inequality is The graph of the solution set on a number line is shown as follows:

36. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 36 Definition of a Rational Inequality A rational inequality is any inequality that can be put into one of the forms where f is a rational function.

37. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 37 Example: Solving a Rational Inequality Solve and graph the solution set: Step 1 Express the inequality so that one side is zero and the other side is a single quotient.

38. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 38 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: Step 1 (cont) Express the inequality so that one side is zero and the other side is a single quotient. is equivalent to It is in the form where

39. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 39 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: We have found that is equivalent to Step 2 Set the numerator and the denominator of f equal to zero. We will use these solutions as boundary points on a number line.

40. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 40 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: Step 3 Locate the boundary points on a number line and separate the line into intervals. The boundary points divide the line into three intervals

41. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 41 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: We have found that is equivalent to Step 4 Choose one test value within each interval and evaluate f at that number. IntervalTest ValueSubstitute intoConclusion –2–2f(x) > 0 for all x in

42. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 42 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: We have found that is equivalent to Step 4 (cont) Choose one test value within each interval and evaluate f at that number. IntervalTest ValueSubstitute intoConclusion 0f(x) < 0 for all x in

43. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 43 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: We have found that is equivalent to Step 4 (cont) Choose one test value within each interval and evaluate f at that number. IntervalTest ValueSubstitute intoConclusion 2f(x) > 0 for all x in

44. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 44 Example: Solving a Rational Inequality (continued) Solve and graph the solution set: Step 5 Write the solution set, selecting the interval or intervals that satisfy the given inequality. The solution set of the given inequality is or The graph of the solution set on a number line is shown as follows:

45. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 45