1. Warm-Up: FACTOR x2 – 36 5x2 - 20 3x + 7 x2 – x – 2 x2 – 5x – 14

2. Answers (x + 6)(x – 6) 5(x + 2)(x – 2) Unfactorable (x – 2) (x + 1)

3. GRAPHING RATIONAL FUNCTIONS
A rational function has a rule given by a fraction whose numerator and denominator are polynomials, and whose denominator is not 0.
EXAMPLES:

4. Characteristics of Rational Functions
Domain
Range
X-intercepts (zeros)
Y-intercept
Interval of Increase
Interval of Decrease
Local Extrema (Minima, Maxima)
Vertical Asymptote
Horizontal Asymptote
Slant Asymptote
Holes

5. An asymptote is a line that a graph approaches more and more closely.
EXAMPLES:
Vertical asymptote

6. Domain
Domain
Set of “input” values
The first coordinate of an ordered pair
(DOMAIN, range)
To find the limitations on the domain of rational functions, set the denominator ≠ 0. These are the value(s) that are NOT in the domain.

7. Find the domain

8. Find the domain

9. Find the domain

10. Finding x-intercepts
Set the numerator= 0.
Factor.
Solve for x.

11. Find the x-intercept

12. Find the domain

13. Find the domain

14. Finding y-intercepts
Plug in ‘0’ for all x-values.
Solve for y.

15. Find the y-intercept

16. Find the y-intercept

17. Find the y-intercept

18. Finding Vertical Asymptotes and Holes in the Graph
1. Factor if possible and cancel identical terms
Set the cancelled term = 0 …this is a hole in the graph
2. Set the remaining term in the denominator = 0 and solve to find the V.A.

19. Example
Hole: x = -1; (-1, 1)
Vertical Asym: x = -2

20. Example
Hole: NONE
Vertical Asym: x = -5/3

21. Example
Hole: x = -2; (-2, 1/2)
Vertical Asym: x = -4

22. Write the equations for the vertical asymptotes.

23. Find the holes and equations for the vertical asymptotes.
No holes.
No holes.
No holes.
No holes.

24. Identifying Degrees of Functions
What is the degree of each function?
2x2 – x – 1
2x – 1
x3 – 5x + 6
x3 – 2x2 + 7x – 1 2 1 3 3

25. Identifying Horizontal Asymptotes
Compare the degrees of the polynomials in the numerator and denominator to find the horizontal asymptotes of a function.

26. If the degree of the numerator and denominator are equal, then
is the H.A.
If the degree of the NUMERATOR is GREATER than the degree of denominator, then there is NO H.A.
If the degree of the NUMERATOR is LESS than the degree of the denominator, then is the H.A.

27. Example

28. Example

29. Example
No Horizontal Asymptote

30. Write the equations for the horizontal asymptotes.

31. Write the equations for the horizontal asymptotes.
No H.A.
No H.A.

32. Slant Asymptotes

33. Slant Asymptotes
If the degree of the numerator is EXACTLY ONE MORE than the degree of the denominator, then the graph has a slant asymptote.

34. How to Find the Equation of a Slant Asymptote
Use long division
Ignore the remainder
The equation that is your answer after dividing is the equation of the slant asymptote.

35. Write the equation of the slant asymptote.
y = x
No Slant. Degrees are equal.
No Slant. Degree of numerator is smaller.
y = x - 2

36. Graphing Rational Functions
Find the VA. Sketch a vertical dotted line for the VA.
Find the HA. Sketch a horizontal dotted line for the HA.
Find the SLA (if any). Sketch a dotted line for the SLA.
Find the x-int by setting the numerator = 0.
Find the y-int by setting x = 0.
Find additional points on both sides of the x value(s) of the VA.

37. In the following problems, a) identify all intercepts, b) find any vertical and horizontal asympotes, c) plot additional solution points, d) sketch the graph.

38. A) Identify all intercepts B) Find all vertical and slant asymptotes C) Plot additional points as needed D) Sketch the graph

39. A) Identify all intercepts B) Find all vertical and slant asymptotes C) Plot additional points as needed D) Sketch the graph

40. A) Identify all intercepts B) Find all vertical and slant asymptotes C) Plot additional points as needed D) Sketch the graph