1. 2.6 Rational Functions

2. A rational function is a function of the form
Where p and q are polynomials functions and q is not the zero polynomial. The domain consists of all real numbers except those for which the denominator q is 0

3. Find the domain of the following rational functions
All Real Numbers

4. Vertical Asymptote: If, as x approaches some number c, the value of then the line x=c is the vertical asymptote of the graph of R C
The vertical asymptote is not part of your graph, it’s a boundary line that the graph gets closer and closer to but does not cross

5. Vertical Asymptotes Rule: A rational function will have a vertical
asymptote at x=r if when r is substituted in for x it makes the denominator zero but not the numerator
f(x) has vertical asymptotes at
In other words, a rational function in lowest terms will have a vertical asymptote at any value of x where the function is undefined

6. Find the vertical asymptotes of the following rational functions
All Real Numbers
None

7. + starting on the right
- starting on the left

8. + starting on the right
- starting on the left

9. + starting on the right
- starting on the left

10. Find vertical asymptotes
Find the domain
Y-max 2
X-min -4
X-max 6
Y-min -2

11. What’s the domain of each of these functions?
Therefore at x=5 there must be a hole.
Graphing utilities do not show holes but if you trace this value it will show no y value
Why isn’t there an asymptote at x=5?
g(x) can’t have a value at x=5
Graph each function

12. Hole Rule: If a rational function has x-a as a factor of the numerator and denominator then the graph will have a hole at a
The graph will have a hole at a. there will be vertical asymptotes at x=d and x=e
The calculator will not show holes but if you trace that value it will be undefined.
Put a circle at x=a where the graph has a hole

13. A vertical asymptote at x=-2 and a hole at x=5
Find any vertical asymptotes or holes on the graph of the following rational functions
A vertical asymptote at x=-2 and a hole at x=5
A vertical asymptote at x=2 and a hole at x=-4
Vertical asymptotes at x=5 and x=-5. There are no holes
No vertical asymptotes and a hole at x=-1

14. P odd, 9-12, 15-18, odd

15. Horizontal Asymptotes: If as or as
Horizontal Asymptotes: If as or as , the values of R(x) approach some fixed number L, then the line y=L is a horizontal asymptote of the graph of R
A horizontal asymptote is not part of the graph. It is a boundary line the graph approaches as the value of x gets farther and farther away from 0. The graph can cross a horizontal asymptote.

16. f(x) has a horizontal asymptote at y=0
Horizontal Asymptote rule #1: If the degree of the numerator is less than the degree of the denominator then there will be a horizontal asymptote at y=0.
Degree 2
Degree 3
f(x) has a horizontal asymptote at y=0

17. f(x) has a horizontal asymptote at y=0
Degree 2
Degree 3
f(x) has a horizontal asymptote at y=0
The reason y=0 is a horizontal asymptote is because when x gets larger and larger the bottom of the fraction gets larger faster than the numerator and the fractions value gets closer and closer to zero x
f(x) 1 10
100
1000
1/-10=-0.1
442/575=0.7687
40492/959795=
A number that is cubed gets larger much faster than one that is squared

18. f(x) has a horizontal asymptote at y=0
Degree 2
Degree 3

19. f(x) has a horizontal asymptote at y=0
Degree 2
Degree 3
f(x) has a horizontal asymptote at y=0

20. Find any asymptotes and holes.

21. Find any asymptotes and holes.

22. Rule #2 for Horizontal Asymptotes: If the degree of the numerator equals the degree of the denominator then there will be a horizontal asymptote at y=a/b if a is the lead coefficient of the numerator and b is the lead coefficient of the denominator.
Horizontal Asymptote at 3 4

23. Is a horizontal asymptote
If I divide f(x) x
f(x) 5 10
100
1000
98/72=1.3611
348/332=1.0482
30498/39212=
For larger and larger values of x this fraction will get closer and closer to zero which makes the value of the function get closer to 3/4

24. Is a horizontal asymptote

25. 3/4
3/4

26. Find any asymptotes or holes

27. Find any asymptotes or holes2 2
3/5
3/5

28. The line y=x is an oblique asymptote
A graph can cross an oblique asymptote but will eventually get closer and closer to it
Oblique Asymptotes : If as or as , the values of R(x) approach some line that is not horizontal, then the line is an oblique asymptote of the graph of R

29. A rational function has an oblique asymptote when the degree of the numerator is one more than the denominator.
f(x) has an oblique asymptote
degree 3
degree 2
To find the oblique asymptote start by dividing so the fraction can be rewritten as a polynomial plus a proper rational function

30. Y=2x-5 is the oblique asymptote.
To find the oblique asymptote start by dividing so the fraction can be rewritten as a polynomial plus a proper rational function
As x gets larger and larger this fraction gets smaller and small but never equals zero. Therefore, the graph approaches the line y=2x-5

31. Y=2x-5 is the oblique asymptote.

32. Y=2x-5 is the oblique asymptote.

33. 1 1 3 3 -2 Find the oblique (slant) asymptote. Degree 2 Degree 1
If the degree of the numerator is one more than the degree of the denominator there will be an oblique asymptote. 1 1 3 3 -2
Don’t use the remainder
The oblique asymptote is

34. -1 2 -2 1 2 Find the oblique (slant) asymptote. Degree 2 Degree 1
If the degree of the numerator is one more than the degree of the denominator there will be an oblique asymptote. -1 2 -2
Don’t use the remainder 1 2
The oblique asymptote is

35. As x gets larger 9/x will approach zero and the graph will approach y=x1 -9
Don’t use the remainder
Oblique asymptote

36. Find the asymptotes

37. Find the asymptotes

38. P377 – #13, 14, 19, 20, 24, 26, 30, 37 – 41, 81 – 87 odd part a
Find any slant or oblique asymptotes
Find the equation of any vertical asymptotes and the x value of any holes.
Find any horizontal asymptotes

39. 7 Steps to Graph a Rational Function. Different than book
1. Find the y-intercept (if there is one) by finding f(0)
2. Factor numerator and denominator if possible
3. Find x-intercepts (if there are any) by setting numerator equal to zero and solving for x.
4. Find any vertical asymptotes (if there are any) and draw on the graph as dotted lines
5. Find any horizontal or oblique asymptotes (if there are any) and draw as dotted lines.
6. Plot on your calculator and trace to get additional points to help with shape.
7. Add in any holes
Make sure your graph does not cross any vertical asymptotes. Make sure your graph approaches horizontal and oblique asymptotes as x approaches positive and negative infinity.

40. Factor the numerator and denominator if possible
Domain
Graph
Find y-intercept
Factor the numerator and denominator if possible
Find x - intercept
A fraction equals zero when the top is zero

41. Domain
Graph
y-intercept = 1/9 x-intercept = 1
Since the fraction will not reduce and the denominator equals zero at 3 and -3 there are vertical asymptotes at x=3 and x=-3

42. y-intercept = 1/9 x-intercept = 1
Graph
y-intercept = 1/9 x-intercept = 1
vertical asymptotes at x=3 and x=-3
The degree of the numerator is less than the degree of the denominator, therefore there is a horizontal asymptote at y=0

43. horizontal asymptote at y=0
Graph
vertical asymptotes at x=3 and x=-3
y-int (0,1/9)
X-int (1,0)
horizontal asymptote at y=0
Now Graph on your calculator.
Use the information you gathered to set your window

44. horizontal asymptote at y=0
Graph
vertical asymptotes at x=3 and x=-3
y-int (0,1/9)
X-int (1,0)
horizontal asymptote at y=0

45. There is a hole at x=-2 and a vertical asymptote at x=2
Graph:
Domain:
There is a hole at x=-2 and a vertical asymptote at x=2
The degree of the numerator equals the degree of the denominator so there is a horizontal asymptote at y=3/2

46. Graph:
Domain:
There is a hole at x=-2 and a vertical asymptote at x=2
There is a horizontal asymptote at y=3/2
If I let the numerator of the reduced fraction equal zero I find there is an x-int at x=3

47. If I let x=0 I find a y-int at (0,9/4)
Graph:
Domain:
There is a hole at x=-2 and a vertical asymptote at x=2
There is a horizontal asymptote at y=3/2
x-int at x=3
If I let x=0 I find a y-int at (0,9/4)

48. Graph:
Domain:
There is a hole at x=-2 and a vertical asymptote at x=2
There is a horizontal asymptote at y=3/2
x-int at x=3
y-int at (0,9/4)
Graph on you calculator and draw in
Make sure to add the hole

49. Graph:
Domain:

50. y=x+4 is the oblique asymptote
Graph:
Domain:
y=x+4 is the oblique asymptote
Vertical Asymptote at x=5 and x=-5
X-intercepts at x=-3,6, & -7
Y-intercept at
The degree of the numerator is one more than the denominator so there is an oblique asymptote. Since I need to divide the polynomial I will multiply the numerator and denominator out then use long division.

51. y=x+4 is the oblique asymptote
Graph:
Domain:
X-intercepts at x=-3,6, & -7
Vertical Asymptote at x=5 and x=-5
y=x+4 is the oblique asymptote
Y-intercept at

52. Graph:

53. Wkst 1
Wkst 2
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