1. Graph Sketching: Asymptotes and Rational Functions
Unit 6 Lesson #2 Asymptotes
Graph Sketching: Asymptotes and Rational Functions
OBJECTIVES
Find limits involving infinity.
Determine the asymptotes of a function’s graph.

2. Unit 6 Lesson #2 Asymptotes
DEFINITION:
A rational function is a function f that can be described by
where P(x) and Q(x) are polynomials, with Q(x) not the zero polynomial. The domain of f consists of all inputs x for which Q(x) ≠ 0.

3. Unit 6 Lesson #2 Asymptotes
A vertical asymptote is a vertical line that a function approaches, but never reaches.
If a is a value for x that makes the denominator = 0, the line x = a is a vertical asymptote.
The graph will never cross a vertical asymptote

4. Unit 6 Lesson #2 Asymptotes
The line x = a is a vertical asymptote if any of the following limit statements are true:
x = a ∞
x = a ∞ -∞

5. Unit 6 Lesson #2 Asymptotes
A horizontal asymptote behaves differently.
We determine what happens to the function as x approaches positive or negative infinity.
The graph of the function can cross the horizontal asymptote.
y = 0

6. Unit 6 Lesson #2 Asymptotes
EXAMPLE 1: Graph the function
x-intercept
y-intercept
Does not exist
does not exist
does not exist
There is a vertical asymptote at x = 1

7. Unit 6 Lesson #2 Asymptotes
EXAMPLE 1: Graph the function
Now lets look at the one-sided limits around x = 1 1 x
1.0001
1.001
1.01
1.2
1.5 2 3 y
30 000
3000
300 15 6 ∞ 1 x -3 -1
0.5
0.9
0.99
0.999 y
-0.75
-1.5 -6
-30
-300
-3 000 -∞

8. Unit 6 Lesson #2 Asymptotes
EXAMPLE 1: Graph the function ∞
y = 0
There is a horizontal
asymptote at y = 0
(0, -3)
x = 1 -∞

9. Unit 6 Lesson #2 Asymptotes
EXAMPLE 2: Graph the function
x-intercept
y-intercept

10. Unit 6 Lesson #2 Asymptotes
EXAMPLE 2: Graph the function
does not exist
does not exist
There is a vertical asymptote at x = 3
x = 3

11. Unit 6 Lesson #2 Asymptotes
Now lets look at the one-sided limits around x = 3 ∞ -∞

12. Unit 6 Lesson #2 Asymptotes
Look for Horizontal Asymptotes ∞
y = 2
x = 3 -∞
There is a horizontal
asymptote at y = 2

13. Unit 6 Lesson #2 Asymptotes
EXAMPLE 2: Graph the function ∞
y = 2
x = 3 -∞

14. EXAMPLE 3: Graph the function
x- intercept
y-intercept
Domain: x ≠ -1, 3

15. Unit 6 Lesson #2 Asymptotes
EXAMPLE 3: cont
There is a 'hole' at (-1, ¾ )

16. Unit 6 Lesson #2 Asymptotes
EXAMPLE 3: cont
DNE
x = 3
There is a vertical asymptote
at x = 3

17. Unit 6 Lesson #2 Asymptotes
Look at one-sided limits around x = 3
From the left
From the right
f(2.999) = ∞
f(3.001) = ∞ ∞
x = 3 -∞

18. Unit 6 Lesson #2 Asymptotes
Is there a horizontal asymptote?
There is a horizontal asymptote at y = 1 ∞
y = 1
x = 3 -∞

19. Unit 6 Lesson #2 Asymptotes∞
x = 3
y = 1
(2, 0)
- ∞