1. Add Holes

2. Section 2.6 Rational Functions Grab out a calc!

3. Exploration Graph using a calculator – what patterns do you notice?

4. Rational Function Where N(x) and D(x) are polynomials and D(x) is not 0

5. Find the domain of f(x)

6. Steps to graph rational functions 1.Intercepts a.y-intercept b.x-intercept(s) 2. Asymptotes a. Vertical (0’s in the denominator) (x = ?) b. Horizontal (case 1, 2, or 3?) (y = ?) 3. Test for symmetry 4. Plot points! 5.Use smooth curves to complete the graph

7. Asymptotes Vertical  at zeros of the denominator Horizontal  N < D asymptote at y = 0  N = D asymptote at y = a N / a D  N > D no Horizontal asymptote

8. Ex 1: Graph 1a. y-intercept when x = 0 f(0) = undefined No y-intercept!

9. Graph 1b. Find the zeros of the numerator (x-intercepts) 0 = x 2 – 4 x = {-2, 2} = (x + 2)(x – 2)

10. Graph 2a. Find the zeros of the Denominator (vertical asymptotes) 0 = x 2 x = {0} x = 0 is a vertical asymptote

11. Graph 2b. Find and sketch the horizontal asymptote Since the degree of the numerator and the denominator are the same the horizontal asymptote will be y = 1/1=1 y = 1

12. Graph 3. Test for Symmetry Since f(x)= f(-x) we know this drawing will be symmetrical about the y-axis

13. Graph 4. Plot more points f(1)= f(-1) = -3 f(3)= f(-3) = 5/9

14. Graph 5. Use smooth curves to complete the graph.

15. Ex 2:Find the domain - try on your own and graph of g(x) Horizontal Asymptote at y = 0 when degree of numerator less than degree of the denominator. Vertical Asymptote at the zero of the denominator. y-int (0, -1/2) No x-int

16.  Ex 3: Graph f(x) No x or y intercepts

17.  Ex 3: Graph f(x) No x or y intercepts

18. Ex 4: Graph Factor Horizontal asymptotes Vertical Asymptotes No x-int y-int (0, -1/6)

19. Ex 4: Graph

22. Ex 5: So what happens if there isn’t a horizontal asymptote??? Graph Lets rewrite the function to get a better idea of what’s happening Approaches 0 as x approaches infinity Thus, we have a slant asymptote! Slant Asymptote

23. Ex 5: So what happens if there isn’t a horizontal asymptote??? Slant Asymptote

24. Ex 6: Graph Lets rewrite the function to get a better idea of what’s happening Approaches 0 as x approaches infinity Thus, we have a slant asymptote! Slant Asymptote

25. Ex 6: Graph Slant Asymptote

26. Ex 7 Graph

27. No asymptote at x = 2, but there is a hole!!!

28. H Dub 2-6 Page 193 #13-20all, 21-33ODD, 38, 41, 45, 53, 56, 59