1. Graphing Rational Functions
Unit 3
Graphing Rational Functions

2. Find the Vertical Asymptotes Find the Horizontal Asymptotes
Objectives:
Part I
Find the Vertical Asymptotes
Find the Horizontal Asymptotes

3. Rational Functions
An asymptote is a vertical or horizontal line where the function is undefined. (Dashed line)
A rational function can have more than one asymptote:
Vertical asymptote (are the “excluded values)
Horizontal asymptote (at most one, must apply the rules for)

4. Vertical Asymptotes Find the vertical asymptote of
V.A. is a vertical line where x = a, and “a “ represents the
real zeros of the denominator ( excluded valued).
Find the vertical asymptote of
Example:
Since the zeros are 1 and -1. Thus the vertical asymptotes are x = 1 and x = -1.
Recall: x cannot be 1 or -1 if you solved it.

5. Horizontal Asymptotes
The horizontal asymptote is determined by looking at the degrees of p(x) and q(x).
**In other words, the degree of the numerator and degree of the denominator**

6. (horizontal asymptote).
Rules for the HA
(horizontal asymptote).
If the degree of p(x) is less than the degree of q(x), then the horizontal asymptote is y = 0.
b. If the degree of p(x) is greater than the degree
of q(x), then there is no horizontal asymptote.
c. If the degree of p(x) is equal to the degree of
q(x), then the horizontal asymptote is a ratio of…

7. Let’s try 1 Find the horizontal asymptote: Example 1:
Degree of numerator = 1
Degree of denominator = 2
y = 0.
Since the degree of the numerator is less than the degree of the denominator, horizontal asymptote is

8. Find the horizontal asymptote:
Example 2:
Find the horizontal asymptote:
Degree of numerator = 1
Degree of denominator = 1
Since the degree of the numerator is equal to the degree of the denominator, horizontal asymptote is

9. Example 3 Find the horizontal asymptote: Example:
Degree of numerator = 2
Degree of denominator = 1
There is no horizontal asymptote. Since the degree of the numerator is greater than the degree of the denominator.

10. Then plot 2 points to the left and right of the center
Copy …..
Graphing a Rational Function
The graph of a has the following characteristics.
Vertical asymptotes:
Horizontal asymptotes:
Then plot 2 points to the left and right of the center

11. Break.  Video on Asymptotes http://cms. gavirtualschool
Break!  Video on Asymptotes Tab 4 , Video 1/3

12. Let’s practice… Handout Determine the H.A and V. A for each

13. Part II: Graphing Rational Function
STEPS!
Identify your asymptotes and sketch them. (Horizontal and Vertical)
Create a table of values to graph at least 2 points on either side of the vertical asymptote.
Sketch!

14. “Hyperbola”- Type of Rational Function.
One form:
2nd Form (Standard form of a Hyperbola:
If using 2nd form
Asymptotes: x=h (vertical) Note: If denominator in equation is (+), equation will be a neg “h”). y=k (horizontal/take whatever sign with it)
Find 2 points on either side of the vertical asymptote. (Evaluate to find (x,y) coordinates.
Graph the asymptotes.
Plot the points and sketch graph.

15. Example 1 Graphing a Rational Function Vertical asymptotes:
Identify your asymptotes and sketch them. (Horizontal and Vertical)
Create a table of values to graph at least 2 points on either side of the vertical asymptote.
Sketch!
Graphing a Rational Function
Graph the function. State the domain and range.
Vertical asymptotes:
Horizontal asymptotes: x y

16. Example 2 . x y -5 1.5 -2 1 2 5 4 3 Vertical Asymptote: x=1
Identify your asymptotes and sketch them. (Horizontal and Vertical)
Create a table of values to graph at least 2 points on either side of the vertical asymptote.
Sketch!
Vertical Asymptote: x=1
Horizontal Asymptote: y=2
x y
Left of vert. asymp.
Right of vert. asymp.

17. Example 3~ You try! Graphing a Rational Function center:
Graph the function.
center:
Vertical asymptotes:
Horizontal asymptotes: x y

18. Example 4 x y -3 .83 -2 1.33 0 -.67 2 0 Vertical asymptote:
3x+3=0 (set denominator =0)
3x=-3
x= -1
Horizontal Asymptote:
x y

19. Think Pair Share- 5-10 Handout from yesterday!

20. Day 3: Domain and Range Monday..
Domain-The domain of each graph is all real numbers except what makes the denominator zero. (X Values)
Range – All y values

21. COPY! When describing the Domain and Range from the Standard equation form OF THE HYPERBOLA. Consider the following!

22. E State the domain & range.
Ex. 1
The Domain & Range is….
Domain: (-∞, 1) U (1, ∞)
Range: (-∞, 2) U (2, ∞)

23. Domain: (- ∞, -2) U (-2, 2) U (2, ∞)
Ex. 2 When describing the domain from the graph itself!
Domain: (- ∞, -2) U (-2, 2) U (2, ∞)
Range: (- ∞, 0] U (3, ∞)

24. Domain: Range: Vertical asymptotes: Horizontal asymptotes: x y
State the domain and range.
Vertical asymptotes:
Horizontal asymptotes: x y
Domain:
Range:

25. Domain: Range: You try! Vertical asymptotes: Horizontal asymptotes: x
State the domain and range.
Vertical asymptotes:
Horizontal asymptotes: x y
Domain:
Range:

26. Guided Practice 5-10 Describe the Domain and Range

27. Characteristics: Discontinuity and Holes
Discontinuity- Where the graph breaks ( The Vertical Asymptotes)
Holes – Factor and simplify. They are the factor/s that will cancel. Solve for the “x”.
Example 4 – Handout 2
HW- Describe all characteristics

29. Ex
You try AGAIN!
Domain: (-∞,∞)
Range:

30. Let’s practice Describe the Domain and Range # 5, 6 from Handout