1. Rational Functions and Asymptotes (Section 2-6)

2. (a) Find the domain of the function, (b) complete each table, and (c) discuss the behavior of f near any excluded values.
Example 1 x
f(x)
-0.5
-0.1
-0.001 x
f(x)
0.5
0.1
0.001
0.0001

3. (a) Find the domain of the function, (b) complete each table, and (c) discuss the behavior of f near any excluded values.
Example 2 x
f(x)
0.5
0.9
0.99
0.999 x
f(x)
1.5
1.1
1.01
1.001

4. Parent Function
pg 147

5. Notation:
f(x) → - ∞ as x → 0- means f(x) approaches -∞ without bound (i.e. goes towards -∞ as x approaches 0 from the left
f(x) → ∞ as x → 0+ means f(x) approaches ∞ without bound (i.e. goes towards ∞ )as x approaches 0 from the right.
f(x) → 0 as x → -∞ means f(x) approaches 0 as x approaches - ∞ without bound (i.e. goes towards -∞)
f(x) → 0 as x → ∞ means f(x) approaches 0 as x approaches ∞ without bound (i.e. goes towards ∞)

6. Asymptotes Defined:
The line x=a is a vertical asymptote of the graph of f if f(x) → ∞ or f(x) → ∞ as x → a from either the left or the right.
The line y = b is a horizontal asymptote of the graph of f if f(x) → b as x → ∞ or x → - ∞
An asymptote is a line (either horizontal or vertical ) that the graph approaches but never touches.

7. Determining Horizontal and Vertical Asymptotes
Vertical: Set the denominator equal to zero and solve
Horizontal:
Compare Degree of Numerator to Degree of Denominator
Horizontal Asymptote
Less than
y = 0
Equal
Greater than
None
LC = Leading Coefficient
Holes: What cancels in denominator

8. pg 147

9. Identify any horizontal and vertical asymptotes.
Example 3

10. Identify any horizontal and vertical asymptotes.
Example 4

11. Identify any horizontal and vertical asymptotes.
Example 5

12. Match the function with its graph.
Example 6 A B C

13. (a) Identify any horizontal and vertical asymptotes and (b)identify any holes in the graph.
Example 7

14. (a) Identify any horizontal and vertical asymptotes and (b)identify any holes in the graph.
Example 8

15. (a) Find the domain of the function (b) decide if the function is continuous (c)identify any horizontal and vertical asymptotes.
Example 9

16. (a) Find the domain of the function (b) decide if the function is continuous (c)identify any horizontal and vertical asymptotes.
Example 10

17. (a) Find the domain of the function (b) decide if the function is continuous (c)identify any horizontal and vertical asymptotes.
Example 11

18. HW #61 pg (1, 3, 7-12all, odd)

19. Find the zeros (if any) of the rational function.
Example 12

20. Find the zeros (if any) of the rational function.
Example 13

21. Find the zeros (if any) of the rational function.
Example 14

22. Find the zeros (if any) of the rational function.
Example 15

23. If you are given characteristics of a function and asked to write a function that would match those characteristics, think about what each piece means for the numerator and the denominator and then put all of the information together.

24. Write a rational function f that has the specified characteristics.
Example 16
Vertical asymptote: x = -1
Horizontal asymptote: y = 0
Zeros: x = 2

25. Write a rational function f that has the specified characteristics.
Example 17
Vertical asymptotes: x = -1, x = 2
Horizontal asymptote: y = -2
Zeros: x = -2, x = 3

26. HW #62 pg 153 (23, 25, all)