This document discusses rational functions and their graphs. It covers key concepts such as finding and graphing vertical and horizontal asymptotes of rational functions. Examples are provided to illustrate how to find asymptotes and graph different types of rational functions, including transformations and slant asymptotes. Applications involving rational functions are also presented.

1. Section 3.5 Rational Functions and Their Graphs

2. Rational Functions

4. Example Find the domain of the rational function.

5. Example Find the domain of the rational function.

6. Vertical Asymptotes of Rational Functions

11. Two Graphs with Vertical Asymptotes, one without

12. Graphing Calculator Connected mode Dot Mode

13. Example Find the vertical asymptote, if any, of the graph of the rational function.

14. Example Find the vertical asymptote, if any, of the graph of the rational function.

15. Example Find the vertical asymptote, if any, of the graph of the rational function.

16. A graph with a hole corresponding to the denominator’s zero. Your calculator will not show the hole.

17. Horizontal Asymptotes of Rational Functions

19. Two Graphs with Horizontal Asymptotes, one without Notice how the horizontal asymptote intersects the graph.

20. Example Find the horizontal asymptote, if any, of the graph of the rational function.

21. Example Find the horizontal asymptote, if any, of the graph of the rational function.

22. Using Transformations to Graph Rational Functions

23. Graphs of Common Rational Functions

24. Transformations of Rational Functions

25. Example

26. Example

27. Graphing Rational Functions

29. Example

30. Example

31. Slant Asymptotes

33. Example

34. Example

35. Applications

37. Example The Fort Myers Fishing Company discovered a better material for making fishing reels. The fixed monthly cost is $10,000 for the cost of rental of space, manufacturing equipment, as well as wages and benefits for it’s employees. It costs $10 for materials to make each fishing reel.

38. (a) (b) (c) (d)

39. (a) (b) (c) (d)

40. (a) (b) (c) (d)