1. Rational Functions and Their Graphs
Section 2.6
Page 326

2. Definitions
Rational Function- a quotient of two polynomial functions in the form
f(x) = p(x) q(x) ≠ 0
q(x)
Domain:

3. Example 1
Find the domain of each rational function

4. Reciprocal Function
Arrow Notation
(see page 328)

5. Arrow Notation

6. Use the graph to answer the following questions.
As x → -2-, f(x) →
As x → -2+, f(x) →
As x → 2-, f(x) →
As x → 2+, f(x) →
As x → -, f(x) →
As x → , f(x) →

7. Vertical Asymptotes
Definition: the line x = a is a vertical asymptote of the graph of a function if f(x) increases or decreases (goes to infinity) without bound as x approaches a
Locating Vertical Asymptotes: set the denominator of your rational function equal to zero and solve for x
Find the vertical asymptotes of f(x) = x – 1
x2 – 4

8. Homework
Page 342 #1 - 28

9. Holes
A value where the denominator of a rational function is equal to zero does not necessarily result in a vertical asymptote.
If the numerator and the denominator of the rational function has a common factor (x – c) then the graph will have a hole at x = c
Example: f(x) = (x2 – 4)
x – 2

10. Finding the Horizontal Asymptote
First identify the degree (highest power) of p(x) and q(x).
f(x) = p(x) degree n
q(x) degree m
and identify their leading coefficients.
n < m
The horizontal asymptote is y = 0
n = m
The horizontal asymptote is the ratio of the leading coefficients
n > m
There is no horizontal asymptote

11. Find the Vertical and Horizontal Asymptotes

12. Review Transformation of Functions
Describe how the graphs of the following functions are transformed from its parent function.

13. Homework
Page 342 #

14. Graphing Rational Functions
Seven Step Strategy – page 334
Check for symmetry
Find the intercepts
Find the asymptotes – check for holes
Plot additional points as necessary

15. Example 6 – Graph
Symmetry
Intercepts
Asymptotes
Plot points

16. Example – Graph
Symmetry
Intercepts
Asymptotes
Plot points

17. Slant Asymptotes
Slant Asymptotes occur when the degree of the numerator of a rational function is exactly one greater than that of the denominator
Note- when the degrees are the same or the denominator has a greater degree the function has a horizontal asymptote.
Line l is a slant asymptote for a function f(x) if the graph of y = f(x) approaches l as x → ∞ or as x → -∞ l

18. Determine the Slant Asymptote
Use synthetic division to find the slant asymptote then graph the function

19. Find the Slant Asymptote
use long division

20. Partner Work
Check for symmetry then find the intercepts, asymptotes, and holes of each rational function

21. Homework
Page 342 #49 – 78 do 2 skip 1