1. Precalculus Section 2.7 2015 Objective: To sketch graphs of rational functions Refer to “Quick Guide to Rational Functions.”

2. HWQ (No Calculator) For the function, find a)Vertical Asymptotes b)Horizontal Asymptotes c)X-intercepts d)Y-intercpets e)Domain

3. y-int. (, ) x-int. (, ) Domain: Asymptote(s) Let x = 0 to find y-int. 0 Let y = 0 to find x-int.(s) none Where is g(x) undefined? V.A. @ x = 2 Compare the exponents. Where Do we have a horizontal asymptote? Deg. of N < Deg. of D is horz. asymptote

4. x = 2 y = 0

5. y-int. (, ) x-int. (, ) Domain: Asymptote(s) none V.A. @ x = 0 H.A. x = 0 y = 2

6. y-int. (, ) x-int. (, ) Domain: Asymptote(s) (x-2)(x+1) 0 V.A. @ x = -1 x = 2 H.A. y = 0 b/c N < D x = -1 x = 2

7. y-int. (, ) x-int. (, ) Domain: Asymptote(s) You try:

8. Slide 4.6 - 8Copyright © 2010 Pearson Education, Inc. Slant, or Oblique, Asymptotes A third type of asymptote, which is neither vertical nor horizontal, occurs when the numerator of a rational function has degree one more than the degree of the denominator. To find a slant asymptote, divide the rational function and ignore the remainder.

9. Ex: Find all asymptotes on the graph of the function:

10. Slide 4.6 - 10Copyright © 2010 Pearson Education, Inc. Slant, or Oblique, Asymptotes The line y = x + 1 is a slant asymptote, or oblique asymptote of the graph of f.

11. y-int. (, ) x-int. (, ) (, ) Domain: Asymptote(s) (x-2)(x+1) 0 2 2 0 -1 0 V.A.x = 1 Slant asym. y = x x = 1 y = x (You Try)

12. y-int. (, ) x-int. (, ) ( ) Domain: Asymptote(s) (You Try) (x)(x-1) 0 1, 0 V.A.x = -1 Slant asym. y = x-2

13. Try This: Write the equation of a rational function that has a vertical asymptote at x = 1, a horizontal asymptote at y = -2, a hole at x = -3, and an x-intercept at x = 0.

14. Homework: Graphing Rational Functions WS Additional Examples

15. y-int. (, ) x-int. ( ) ( ) Domain: Asymptote(s) (You Try) (x)(x-2)(x+3) 0 0, 0 -3, 0 V.A. x = -1 Slant asym. y = x+2 (x-2)(x+1) Hole @ x=2

16. y-int. (, ) x-int. (, ) (, ) Domain: Asymptote(s) (x-3)(x+3) (x-2)(x+2) V.A. x = -2 x = 2 H.A. 3 0 -3 0 (You Try)

17. y-int. (, ) x-int. (, ) Domain: Asymptote(s) 0 -1 1 0 V.A. x = -1 H.A. (If Time Permits)

18. y-int. (, ) x-int. (, ) (, ) Domain: Asymptote(s) You try

19. y-int. (, ) x-int. (, ) (, ) Domain: Asymptote(s)

20. y-int. (, ) x-int. (, ) (, ) Domain: Asymptote(s)

21. y-int. (, ) x-int. (, ) (, ) Domain: Asymptote(s)

22. y-int. (, ) x-int. (, ) (, ) Domain: Asymptote(s)

23. y-int. (, ) x-int. (, ) (, ) Domain: Asymptote(s)

24. y-int. (, ) x-int. (, ) (, ) Domain: Asymptote(s)