1. Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:

2. Announcements Assignment ◦p. 278 ◦# 3-12, 23, 26 ◦Study Guide Notebook Quiz Wednesday Review Session Tuesday after school

3. 2.7 Rational Functions How to find the domains of rational functions How to find horizontal and vertical asymptotes of graphs of rational functions

4. Introduction to Rational Functions A rational function is a function of the form f(x) = N(x)/D(x), where N and D are both polynomials. The domain of f is all x such that D(x)  0. ◦Example:

5. Example 1. Find the domain of All Reals ≠ -2, 2

6. Horizontal and Vertical Asymptotes

7. Vertical Asymptote X – values where there are no y – values Find vertical asymptotes by finding the domain

8. Horizontal asymptotes The graph of f has one horizontal asymptote or no horizontal asymptote, depending on the degree of n and m. a. If n < m, then y = 0 is the horizontal asymptote of the graph of f. b. If n = m, then y = a n /b m is the horizontal asymptote of the graph of f. c. If n > m, then there is no horizontal asymptote of the graph of f.

9. Hint, hint, note, note Graphs CAN touch a horizontal asymptote Graphs CAN’T touch a vertical asymptote Example

10. Horizontal Asymptote a. If n < m, then y = 0 is the horizontal asymptote of the graph of f.

11. Horizontal Asymptote b. If n = m, then y = a n /b m is the horizontal asymptote of the graph of f.

12. Horizontal Asymptote c. If n > m, then there is no horizontal asymptote of the graph of f.

13. Find any horizontal and vertical asymptotes of the following. The horizontal asymptote is at y= 1/2, and the vertical asymptote is at x = 3/2. What x-values will make the function undefined? What is the relationship between the highest powers in the numerator and denominator?

14. Find any horizontal and vertical asymptotes of the following. No horizontal asymptote and a vertical asymptote at x = -1 What x-values will make the function undefined? What is the relationship between the highest powers in the numerator and denominator?

15. Domain of a rational function To find the domain of a rational function of x,.. set the denominator of the rational function equal to zero and solve for x. These values of x must be excluded from the domain of the function.

16. Warm Up Find the domain of the function and identify any horizontal and vertical asymptotes.

17. Announcements Assignment ◦p. 281 ◦# 69 – 74 ◦Study Guide Notebook Quiz tomorrow Review Session today after school

18. Objectives How to analyze and sketch graphs of rational functions How to sketch graphs of rational functions that have slant asymptotes

19. Steps for finding the Graph of a Rational Functions

20. 1 st Guideline for graphing rational functions 1. Find and plot the y-intercept (if any) by evaluating f(0)

21. 2 nd Guideline for graphing rational functions 1. Find the zeros of the numerator (if any) by setting the numerator = 0. Then plot them as x – intercepts

22. 3 rd Guideline for graphing rational functions 1. Find the zeros of the denominator (if any) by setting the denominator = 0. Then sketch the corresponding vertical asymptotes

23. 4 th Guideline for graphing rational functions 1. Find and sketch the horizontal asymptote (if any) by using the rules for finding the horizontal asymptote

24. 5 th Guideline for graphing rational functions 1. Plot at least one point between and at least one point beyond each x intercept and vertical asymptote XY 3 (1/4)-2 11 X – int. = (1/2) Vert. Asym. = 0

25. 6 th Guideline for graphing rational functions 1. Use smooth curves to complete the graph between and beyond the vertical asymptotes

26. Example 1. Sketch the graph of the following function. y-Intercept:None x-Intercept:(-1, 0) Vertical asymptote:x = 0 Horizontal asymptote:y = 1 Additional points:(-2, 0.5), (-1.5, 1/3), (1, 2)

27. Sketch the graph of each of the following functions. y-Intercept:(0, 0) x-Intercept:(0, 0) Vertical asymptote:none Horizontal asymptote:y = 0 Additional points:(-2,-0.4), (-1, -1/2), (1, 1/2)

28. Slant Asymptotes Slant Asymptotes y = -3x – 3 Is our slant asymptote If n is exactly one more than m, then the graph of f has a slant asymptote at y = q(x), where q(x) is the quotient from the division algorithm.

29. Decide whether each of the following rational functions has a slant asymptote. If so, find the equation of the slant asymptote. (a) Yes, y = x  3 (b) No

30. Example 2. Sketch the graph of y-Intercept:(0, 0) x-Intercept:(0, 0) Vertical asymptote:x = 2 Slant asymptote:y = x + 2 Additional points:(-1/2,-0.1), (1, -1), (3, 9)

31. Slant Asymptotes If n is exactly one more than m, then the graph of f has a slant asymptote at y = q(x), where q(x) is the quotient from the division algorithm.

32. Sketch the graph of each of the following functions. y-Intercept:(0, -0.25) x-Intercept:(2, 0) Vertical asymptote:x = -2 and x = 4 Horizontal asymptote:y = 0 Additional points:(-4, -0.375), (0, 1/4), (3, -1/5), (6, 1/4)

33. Example 2. Find any horizontal and vertical asymptotes of the following. The horizontal asymptote is y = 0. The only vertical asymptote is x = 1. There will be a hole in the graph at x = -1.

34. 1 st Guideline for graphing rational functions

35. 2 nd Guideline for graphing rational functions

36. 3 rd Guideline for graphing rational functions

37. 4 th Guideline for graphing rational functions

38. 5 th Guideline for graphing rational functions XY 3 (1/4)-2 11 X – int. = (1/2) Vert. Asym. = 0

39. 6 th Guideline for graphing rational functions