1. Warm UP!
Factor the following:

2. Rational Functions LG 7-1: Characteristics & Graphs of Rational
(quiz 12/5)
LG 7-2: Inverses of Rational Functions
(quiz 12/12)

3. Rational Functions General Equation:
S and T are polynomial functions
Verbally: f(x) is a rational function of x.
Features: A rational function has a discontinuities - asymptotes and/or holes.

4. The parent rational function is:
The shape is made by the behavior of a function as it approaches asymptotes.
ALL rational functions will look similar to this parent graph as they are all part of the same family.
An example of a rational function is:

5. How to find the Characteristics of Rational Functions
Domain & Range
Intercepts
Discontinuities - Vertical Asymptotes, Horizontal Asymptotes, Slant/Oblique Asymptotes, and Holes

6. All rational functions have asymptotes in their graphs.

7. X-intercepts Where the function crosses the x-axis.
A function can have none, one, or multiples
To find the x-int of a Rational Function, set the numerator equal to zero and solve for x.

8. Find all x-intercepts of each function.
Practice
Find all x-intercepts of each function.

9. y-intercepts Where the function crosses the y-axis
A function can have NO MORE THAN 1!
To find the y-int of Rational Functions, substitute 0 for x.

10. Find all y-intercepts of each function.
Practice
Find all y-intercepts of each function.

11. Vertical Asymptotes
A vertical asymptote is an invisible line that the graph will NEVER cross.
The function is undefined at a VA.
You can find the VA by setting the denominator equal to zero and solving for x.

12. Domain of a Rational Function
The domain of a rational function is all real numbers excluding the discontinuities (the vertical asymptote) and the x-value of the hole (more tomorrow!)

13. The functions p and q are polynomials.
Defn:
Rational Function
The functions p and q are polynomials.
The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero.

14. Domain of a Rational Function
{x | x  –4} or
(-, -4)  (-4, )

15. Domain of a Rational Function
{x | x  2} or
(-, 2)  (2, )

16. Domain of a Rational Function
{x | x  –3, 3} or
(-, -3)  (-3, 3)  (3, )

17. When you have one vertical asymptote…
Your graph is separated into 2 sections…
To the left of the asymptote
To the right of the asymptote

18. When you have two vertical asymptotes…
Your graph is separated into 3 sections…
To the left of both asymptotes
In between the asymptotes
To the right of both asymptotes

19. Find the Vertical Asymptotes & Domain:
Practice
Find the Vertical Asymptotes & Domain:

20. Horizontal Asymptotes of Rational Functions
3.6: Rational Functions and Their Graphs
Horizontal Asymptotes of Rational Functions
If the graph of a rational function has a horizontal asymptote, it can be located by using the following theorem.
Locating Horizontal Asymptotes
Let f be the rational function given by
The degree of the numerator is n. The degree of the denominator is m.
If n < m, the x-axis is the horizontal asymptote of the graph of f.
If n = m, the line y = is the horizontal asymptote of the graph of f.
If n > m, the graph of f has no horizontal asymptote.

21. A horizontal asymptote exists at y = 5/2.
Asymptotes
Horizontal Asymptote
Example
A horizontal asymptote exists at y = 5/2.
A horizontal asymptote exists at y = 0.

22. Find the Horizontal Asymptotes & Range:
Practice
Find the Horizontal Asymptotes & Range:

23. How do you find asymptotes in rational functions?
Vertical Asymptotes
Horizontal Asymptotes
Compare the degrees of the numerator and the denominator
1. Set denominator equal to zero.
1. If bigger on BOTTOM then there is a HA at y = 0.
2. Solve for x
2. If n = m, then there is a HA at y = ___ (find by divide the leading coefficients).
3. If bigger on TOP, then there is no horizontal asymptotes (this means there could be a SLANT/OBLIQUE asymptote which we will discuss tomorrow!

24. Find all asymptotes of
Vertical:
x = -1
and x = 2
Horizontal:
Degree of top = 1
Degree of bottom = 2
BIGGER on BOTTOM
y = 0

25. Degree of denominator = 1
Find all asymptotes of
Vertical:
x = 0
Horizontal:
Degree of numerator = 1
Degree of denominator = 1
EQUAL

26. No horizontal asymptote
Find all asymptotes of
Vertical:
x - 1 = 0
x = 1
Degree of numerator = 2
Horizontal:
Degree of denominator = 1
Bigger on TOP
No horizontal asymptote

27. Find the following characteristics for the rational function below:
For example:
Find the following characteristics for the rational function below:
Domain:
Range:
x-intercepts:
y-intercepts:
Horizontal asymptote:
Vertical asymptotes:

28. Ex. 4 Find all asymptotes of and graph Vertical: x + 1 = 0 x = -1
Set denominator equal to zero
x = -1
Horizontal:
n > m by exactly one
n = 2
m = 1
No horizontal asymptote
Compare degrees
Slant:
Use long division

29. What are your questions?