1. Section 2.6 Rational Functions Hand out Rational Functions Sheet!

2. HWQ Write the equation of a polynomial with zeros at 3, -1, 4, and 2i.

3. Section 2.6 Rational Functions Objective: To find asymptotes and domain of rational functions.

4. Warm-Up A ball is thrown into the air. It’s height is given by the function Find the maximum height reached by the ball.

5. The ratio of two polynomial functions is called a rational function. Examples Denominator cannot be zero!

6. A rational function is usually written in the form, where p(x) and q(x) have no common factors. Zeros of a rational function: same zeros as p(x). (a rational expression = 0 where its numerator = 0)

7. What is an asymptote? An asymptote is a line that a graph approaches as it moves away from the origin. Asymptotes can be vertical or horizontal. Vertical asymptotes cannot ever be crossed, but horizontal ones can. Vertical asymptotes are determined by the zeros of the denominator. Horizontal asymptotes are determined by comparing the degrees of the numerator vs. denominator.

8. ASYMPTOTE RULES Vertical Asymptotes: located at zeros of q(x). Horizontal Asymptotes: (at most one) a. If degree of the numerator degree of the denominator no horizontal asymptote Slant Asymptote: only if the degree of the numerator is exactly one more than the degree of denominator. Divide numerator by denominator. y = ax+b is the slant asymptote. **In order to get a good sketch of the graph, you must plot some points between and beyond each x-intercept and vertical asymptote.

9. Copyright © 2010 Pearson Education, Inc. Example 4 For each rational function, determine any horizontal or vertical asymptotes. a) b) c)

10. Copyright © 2010 Pearson Education, Inc. Example Solution a) HA: Degree of numerator and denominator are both 1. Since the ratio of the leading coefficients is 6/3, the horizontal asymptote is y = 2. VA: When x = –1, the denominator, 3x + 3, equals 0 and the numerator, 6x – 1 does not equal 0, so the vertical asymptote is x =  1.

11. Copyright © 2010 Pearson Education, Inc. Example Solution continued a) Here’s a graph of f(x).

12. Copyright © 2010 Pearson Education, Inc. Example Solution continued b) HA: Degree of numerator is one less than the degree of the denominator so the x-axis, or y = 0, is a horizontal asymptote. VA: When x = ±2, the denominator, x 2 – 4, equals 0 and the numerator, x + 1 does not equal 0, so the vertical asymptotes are x =  2 and x = 2.

13. Copyright © 2010 Pearson Education, Inc. Example Solution continued b) Here’s a graph of g(x).

14. Copyright © 2010 Pearson Education, Inc. Example Solution c) HA: Degree of numerator is greater than the degree of the denominator so there are no horizontal asymptotes. VA: When x = –1, both the numerator and denominator equal 0 so the expression is not in lowest terms: g(x) = x – 1, x ≠ –1. There are no vertical asymptotes.

15. Copyright © 2010 Pearson Education, Inc. Example Solution c) Here’s the graph of h(x). A straight line with the point (–1, –2) missing. Why isn’t there a vertical asymptote at x=-1?

16. Graph: x = 0 y = 0

17. Graph: x = 0 y = 0 DOMAIN RANGE Vertical Asymptotes? Horizontal Asymptotes? X and y intercepts?None y=0 x=0

18. Graph x y Vertical asymptote at x = 4 Horizontal asymptote at y = 0 x intercept? y intercept? xf(x)f(x) 5 2 0-1/2 2 None

19. x y Graph Vertical asymptote ? Horizontal asymptote ? X intercepts? Y intercepts? Domain? x = 2 and x = -3 y = 0 (0, -1) None Evaluate at convenient values of x.

20. x y Graph Factor: Domain:

21. Analyze the function. Find the following: Vertical asymptotes: horizontal asymptotes: x intercepts: y intercepts: Domain: x y

22. Analyze the function. Find the following: Vertical asymptotes: horizontal asymptotes: x intercepts (reduce 1 st ): y intercepts: Domain:

23. Analyze the graph of: X intercepts? Vertical asymptote ? Horizontal asymptote ? Domain? Range? Y intercepts?

24. x y Graph x f(x)f(x) -3 -2 0 1 2 3 4 5 6.1.2.5 2 und. 2.5.2.1 Vertical asymptote at x = 2 Horizontal asymptote at y = 0

25. Homework P.148 7-19 all, 31-34 all, 35, 39 (table of values not required)