Consider the rational function below.

1. Consider the rational function below. We know that since d = n, f has a horizontal asymptote at y = 2. Since a rational function is telling us to divide, let’s do so. − −

2. Consider the rational function below. We know that since d = n, f has a horizontal asymptote at y = 2. Sof (x) can be rewritten as: Approaches 0 as x → ∞ And our graph is trying to look like y = 2 at large values of x.

3. Consider the rational function below. We know that since d < n, f has no horizontal asymptotes. Since a rational function is telling us to divide, let’s do so. − −

4. Consider the rational function below. • We know that since d < n, f has no horizontal asymptotes. Sof (x) can be rewritten as: Approaches 0 as x → ∞ And our graph is trying to look like y = 2x at large values of x. This is called a slant asymptote.

5. Slant Asymptotes A rational function has a slant asymptote if n = d + 1 • The degree of the numerator is one more than the degree of the denominator To find the equation of a slant asymptote, use long division and forget about the remainder. • At large values of x, the remainder approaches 0 anyway.

6. Exercise 1 Find all asymptotes of the rational function.

7. To Graph: To graph a rational function: • Factor N(x) and D(x). • Find vertical asymptotes (where D(x) = 0) and plot as dashed lines. • If a factor cancels, it is not an asymptote (A Hole) • Find horizontal asymptote (comparing d and n) and plot as a dashed line. • Find slant asymptote (by long division w/o the remainder) and plot as a dashed line. • Plot x- and y-intercepts. • If a factor cancels, it is not a zero (A Hole) • Use smooth curves to finish the graph.

8. More on Asymptotes Vertical Asymptotes: • Your graph can never cross one! • If x = a is a vertical asymptote, then interesting things happen really close to a: • f (x) could approach +∞ or −∞ • Think of vertical asymptotes as black holes that attract values near a

9. More on Asymptotes Vertical Asymptotes: • The end behavior around a vertical asymptote is similar to that of polynomials: V.A. at x = 1 (multiplicity of 1) V.A. at x = 1 (multiplicity of 1)

10. More on Asymptotes Vertical Asymptotes: • The end behavior around a vertical asymptote is similar to that of polynomials: V.A. at x = 1 (multiplicity of 2) V.A. at x = 1 (multiplicity of 2)

11. More on Asymptotes Horizontal Asymptotes: • Your graph can cross one! • Attracts values approaching +∞ or −∞

12. More on Asymptotes Slant Asymptotes: • Your graph can cross one of these, too! • Attracts values approaching +∞ or −∞

13. Exercise 2 Graph:

14. Exercise 3 Graph:

15. Exercise 4 Graph:

16. Exercise 5 Graph:

17. Exercise 6 Graph:

18. Exercise 7 Graph:

19. Exercise 8 Graph: