1. 2.5 – Rational Functions

2. Ex. 1 Graph 5 x – 2

3. Ex. 1 Graph 5 x – 2

4. Ex. 1 Graph 5 x – 2 x – 2 = 0

5. Ex. 1 Graph 5 x – 2 x – 2 = 0 x = 2

6. Ex. 1 Graph 5 x – 2 x – 2 = 0 x = 2 (vertical asymptote)

7. Ex. 1 Graph 5 x – 2 x – 2 = 0 x = 2 (asymptote)

8. Ex. 1 Graph 5 x – 2 x – 2 = 0 x = 2 (asymptote)

9. Ex. 1 Graph 5 x – 2 x – 2 = 0 x = 2 (asymptote) *Graph on Calc.

10. Ex. 1 Graph 5 x – 2 x – 2 = 0 x = 2 (asymptote) *Graph on Calc. Type: y = 5/(x – 2)

11. Ex. 1 Graph 5 x – 2 x – 2 = 0 x = 2 (asymptote) *Graph on Calc. Type: y = 5/(x – 2) 2 nd Table, 3pts on each curve

12. Ex. 1 Graph 5 x – 2 x – 2 = 0 x = 2 (asymptote) *Graph on Calc. Type: y = 5/(x – 2) 2 nd Table, 3pts on each curve

13. Ex. 2 Graph x + 1 x 2 + 3x + 2

14. Ex. 2 Graph x + 1 x 2 + 3x + 2 x + 1 (x + 2)(x + 1)

15. Ex. 2 Graph x + 1 x 2 + 3x + 2 x + 1 (x + 2)(x + 1) 1 x + 2

16. Ex. 2 Graph x + 1 x 2 + 3x + 2 x + 1 (x + 2)(x + 1) 1 x + 2 Asymp. @ x = -2

17. Ex. 2 Graph x + 1 x 2 + 3x + 2 x + 1 (x + 2)(x + 1) 1 x + 2 Asymp. @ x = -2 Hole @ x = -1

18. Ex. 2 Graph x + 1 x 2 + 3x + 2 x + 1 (x + 2)(x + 1) 1 x + 2 Asymp. @ x = -2 Hole @ x = -1 Graph y = (x+1)/(x 2 +3x+2)

19. Ex. 2 Graph x + 1 x 2 + 3x + 2 x + 1 (x + 2)(x + 1) 1 x + 2 Asymp. @ x = -2 Hole @ x = -1 *Graph y = (x+1)/(x 2 +3x+2) *Then 2 nd Table for 3 ordered pairs per curve.

20. Ex. 2 Graph x + 1 x 2 + 3x + 2 x + 1 (x + 2)(x + 1) 1 x + 2 Asymp. @ x = -2 Hole @ x = -1 *Graph y = (x+1)/(x 2 +3x+2) *Then 2 nd Table for 3 ordered pairs per curve.

21. Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain. f(x) = 3x 2 – 3 x 2 – 9

22. Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain. f(x) = 3x 2 – 3 = 3(x + 1)(x – 1) x 2 – 9 (x + 3)(x – 3)

23. Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain. f(x) = 3x 2 – 3 = 3(x + 1)(x – 1) x 2 – 9 (x + 3)(x – 3) Vertical Asymptotes at x = -3 & x = 3

24. Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain. f(x) = 3x 2 – 3 = 3(x + 1)(x – 1) x 2 – 9 (x + 3)(x – 3) Vertical Asymptotes at x = -3 & x = 3 Graph, use Table to find limits for Horizontal.

25. Ex. 3 Determine any asymptotes and intercepts. Then graph and find the domain. f(x) = 3x 2 – 3 = 3(x + 1)(x – 1) x 2 – 9 (x + 3)(x – 3) Vertical Asymptotes at x = -3 & x = 3 Graph, use Table to find limits for Horizontal. Limits show Horizontal Asymptote at y = 3.

26. Domain: {x | x ≠ -3, x ≠ 3}

27. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3

28. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3

29. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3 2(p – 2)

30. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3 2(p – 2) =

31. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3 2(p – 2) = 3p

32. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3 2(p – 2) = 3p 2p – 4 = 3p

33. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3 2(p – 2) = 3p 2p – 4 = 3p -2p -2p

34. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3 2(p – 2) = 3p 2p – 4 = 3p -2p -2p -4

35. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3 2(p – 2) = 3p 2p – 4 = 3p -2p -2p -4 =

36. Proportions: 1 fraction = 1 fraction Ex. 1 Solve each equation. a. p _ = 2 p – 2 3 2(p – 2) = 3p 2p – 4 = 3p -2p -2p -4 = p

37. b. w + w = 4w – 3 w – 1 w – 1

38. b. (w-1)w + w (w-1) = 4w – 3 (w-1) w – 1 w – 1

39. b. (w-1)w + w (w-1) = 4w – 3 (w-1) w – 1 w – 1 w + w(w – 1) = 4w – 3 w + w 2 – w = 4w - 3

40. b. (w-1)w + w (w-1) = 4w – 3 (w-1) w – 1 w – 1 w + w(w – 1) = 4w – 3 w + w 2 – w = 4w – 3 w 2 = 4w – 3

41. b. (w-1)w + w (w-1) = 4w – 3 (w-1) w – 1 w – 1 w + w(w – 1) = 4w – 3 w + w 2 – w = 4w – 3 w 2 = 4w – 3 w 2 – 4w + 3 = 0

42. b. (w-1)w + w (w-1) = 4w – 3 (w-1) w – 1 w – 1 w + w(w – 1) = 4w – 3 w + w 2 – w = 4w – 3 w 2 = 4w – 3 w 2 – 4w + 3 = 0 (w – 3)(w – 1) = 0

43. b. (w-1)w + w (w-1) = 4w – 3 (w-1) w – 1 w – 1 w + w(w – 1) = 4w – 3 w + w 2 – w = 4w – 3 w 2 = 4w – 3 w 2 – 4w + 3 = 0 (w – 3)(w – 1) = 0 w = 3, w = 1

44. b. (w-1)w + w (w-1) = 4w – 3 (w-1) w – 1 w – 1 w + w(w – 1) = 4w – 3 w + w 2 – w = 4w – 3 w 2 = 4w – 3 w 2 – 4w + 3 = 0 (w – 3)(w – 1) = 0 w = 3, w = 1