1. Horizontal Vertical Slant and Holes
ASYMPTOTES TUTORIAL
Horizontal
Vertical
Slant
and Holes
Do Now: graph the following
and name the asymptotes:

2. Definition of an asymptote
An asymptote is a straight line which acts as a boundary for the graph of a function.
When a function has an asymptote (and not all functions have them) the function gets closer and closer to the asymptote as the input value to the function approaches either a specific value a or positive or negative infinity.
The functions most likely to have asymptotes are rational functions

3. Vertical Asymptotes Vertical asymptotes occur when the
following condition is met:
The denominator of the simplified
rational function is equal to 0.
Remember, the simplified rational
function has cancelled any factors
common to both the numerator and
denominator.

4. Finding Vertical Asymptotes Example 1
Given the function
The first step is to cancel any factors common to both numerator and denominator. In this case there are none.
The second step is to see where the denominator of the simplified function equals 0.

5. Finding Vertical Asymptotes Example 1 Con’t.
The vertical line x = -1 is the only vertical asymptote for the function. As the input value x to this function gets closer and closer to -1 the function itself looks and acts more and more like the vertical line
x = -1.

6. Graph of Example 1
The vertical dotted line at x = –1 is the vertical asymptote.

7. x approaches a from the right x approaches a from the left
Rational Functions and Asymptotes
Arrow notation:
Meaning:
x approaches a from the right
x approaches a from the left
x is approaching infinity, increasing forever
x is approaching infinity, decreasing forever ,

8. Vertical Asymptotes
The line x = a is a vertical asymptote of a function if f(x) increases or decreases without bound as x approaches a. If
Then:

9. Graph of Example 1
The vertical dotted line at x = –1 is the vertical asymptote.

10. Finding Vertical Asymptotes Example 2If
First simplify the
function. Factor
both numerator
and denominator
and cancel any
common factors.
*There is no asymptote at x = -3!

11. Finding Vertical Asymptotes Example 2 Con’t.
The asymptote(s) occur where the simplified denominator equals 0.
The vertical line x=3 is the only vertical asymptote for this function.
As the input value x to this function gets closer and closer to 3 the function itself looks more and more like the vertical line x=3.

12. Graph of Example 2 The vertical dotted line at
x = 3 is the vertical asymptote

13. Finding Vertical Asymptotes Example 3If
Factor both the
numerator and
denominator and cancel
any common factors.

14. Finding Vertical Asymptotes Example 3If
Factor both the
numerator and
denominator and cancel
any common factors.
In this case there are no
common factors to
cancel.

15. Finding Vertical Asymptotes Example 3 Con’t.
The denominator equals zero whenever either or
This function has two vertical asymptotes, one at x = -2 and the other at x = 3

16. Graph of Example 3 The two vertical dotted lines at
x = -2 and x = 3 are the vertical asymptotes
Write 2 limit notations for each one.

17. Horizontal asymptotes occur when either one of the following conditions is met (you should notice that both conditions cannot be true for the same function).
Horizontal asymptotes guide the end behavior of graph.
If the degree of the numerator < degree of the denominator then the horizontal asymptote will be y= 0.
If the degree of the numerator = degree of denominator then the horizontal asymptote will be
If the degree of the numerator > degree of the denominator then there is no horizontal asymptote.

18. Find the horizontal asymptote:
Exponents are the same; divide the coefficients
Bigger on Top; None
Bigger on Bottom; y=0

19. Finding Horizontal Asymptotes Example 4If
then there is a horizontal asymptote at the line y=0 because the degree of the numerator (2) is less than the degree of the denominator (3). This means that as x gets larger and larger in both the positive and negative directions (x → ∞ and x → -∞)
the function itself looks more and more like the horizontal line y = 0

20. Graph of Example 4
The horizontal line y = 0 is the horizontal asymptote.

21. Finding Horizontal Asymptotes Example 5If
then because the degree of the numerator (2) is equal to the degree of the denominator (2) there is a horizontal asymptote at the line y=6/5. Note, 6 is the leading coefficient of the numerator and 5 is the leading coefficient of the denominator. As x→∞ and as x→-∞ g(x) looks more and more like the line y=6/5

22. Graph of Example 5
The horizontal dotted line at y = 6/5 is the horizontal asymptote.

23. Finding End behavior Asymptotes Do Now: graph:If
There are no horizontal asymptotes because the degree of the numerator is greater than the degree of the denominator.

24. Graph of previous function

25. Slant Asymptotes
Slant asymptotes occur when the degree of the numerator is exactly one bigger than the degree of the denominator. In this case a slanted line (not horizontal and not vertical) is the function’s asymptote.
To find the equation of the asymptote we need to using synthetic division, (if possible) or long division – dividing the numerator by the denominator.

26. Finding slant asymptotes with synthetic division:
This one has a linear denominator so we can use
synthetic division

27. Note: slants and horizontals don’t happen together!

28. Finding slant asymptotes with synthetic division is only possible when the denominator is linear!
(Remainder was 19)

29. End behavior Asymptotes
Graph the following function:
The end behavior asymptote will be a quadratic
function because the numerator is 2 degrees
larger in the numerator
Use synthetic division to find it.

30. End behavior Asymptotes
Graph the following function:
Note that the remainders don’t matter here!

31. Using long division
Find the quotient and remainder:

32. The slant asymptote
We divide: note that here we will put the first answer over the 5x.

33. The slant asymptote
We divide and find it…. )

34. The slant asymptote We divide and find it….
Y = -2x is the slant asymptote!
7x-9 is just a remainder )

35. Graph of Example 6
Y=-2x

36. Finding a Slant Asymptote Example 7If
There will be a slant asymptote because the degree of the numerator (3) is one bigger than the degree of the denominator (2).
Using long division, divide the numerator by the denominator.

37. Finding a Slant Asymptote Example 7 Con’t.

38. Finding a Slant Asymptote Example 7 Con’t.
We can ignore the remainder
The answer we are looking for is the quotient
and the equation of the slant asymptote is:

39. Graph of Example 7
The slanted line
y = x + 3 is the slant asymptote

40. Finding slant asymptotes with synthetic division:
divide using synthetic division

41. Finding slant asymptotes with synthetic division:
divide using long division

42. Finding slant asymptotes with synthetic division is only possible when the denominator is linear!
(Remainder was 19)

43. Finding Intercepts of Rational Functions
x-intercept: solve f (x) = 0
y-intercept: evaluate f (0)
Do now: find the x and y intercepts:
y=2x+8

44. Intercepts of Rational Functions
Example
x-intercept: solve f (x) = 0
Setting the function = 0 is essentially setting the
numerator= to 0!

45. Intercepts of Rational Functions
y-intercept: evaluate f (0)

46. Find all asymptotes and intercepts
Example Graph
Vertical Asymptote:
Horizontal Asymptote:
x-intercept:
y-intercept:

47. Find the asymptotes and intercepts
Example
Solution Vertical Asymptote:
Horizontal Asymptote:
x-intercept:
y-intercept:

48. Graph all the asymptotes and intercepts
x=3, y=2 (-.5,0) (0, -1/3)

49. Graph all the asymptotes and intercepts
x=3, y=2 (-.5,0) (0, -1/3)
But find a point for the
missing part-try f(6)

50. Graph all the asymptotes and intercepts
x=3, y=2 (-.5,0) (0, -1/3)
But find a point for the
missing part-try f(6)

51. End behavior:
Use long division and divide to find the slant asymptote:

52. Do Now: Use long division and divide: Slant asymptote: Y = x + 1
remainder

53. Quadratic and cubic end behavior asymptotes….
Use long division to find the quotient:

54. Quadratic and cubic end behavior asymptotes….
Use long division to find the quotient:
Quadratic end behavior asymptote:
(Note: the remainder was 11x-43)

55. do now:find the asymptotes
Holes occur in the graph of a rational function whenever the numerator and denominator have common factors. The holes occur at the x value(s) that make the common factors equal to 0.
The hole is known as a removable discontinuity.
When you graph the function on your calculator you won’t be able to see the hole but the function is still discontinuous.

56. Finding a Hole
We were able to cancel the (x + 3) in the numerator and denominator before finding the vertical asymptote.
Because (x + 3) is a common factor there will be a hole at the point where
Finding the coordinates of the hole: substitute -3 into the remaining factors:

57. Finding a Hole There is a hole at
Where are the intercepts and asymptotes?

58. Finding a Hole is a hole (-2,0),and are intercepts
x=3, y=2 are the asymptotes
Sketch the function

59. Graph of Example
Notice there is a hole in the graph at the point where x = -3. You would not be able to see this hole if you graphed the curve on your calculator (but it’s there just the same.)

60. Finding a Hole Use the rational root theorem to factor the numerator
Factor both numerator and denominator to see if there are any common factors.
Find all the intercepts, holes and asymptotes to
Sketch the function

61. Finding a Hole
Factor both numerator and denominator to see if there are any common factors.
there will be a hole at (2,3) asymptotes are x=-2
And a slant at y = x (long division) and an intercept at (0,2). No x intercepts.

62. Graph of last example
There is a hole in the curve at the point where x = 2. This curve also has a vertical asymptote at
x = -2 and a slant asymptote y = x.

63. Problems
Find the vertical asymptotes, horizontal asymptotes, slant asymptotes and holes and intercepts for each of the following functions.
Not all will be found for each. Graph g(x)
Vertical:
Horizontal :
Slant:
Hole:
Vertical:
Horizontal :
Slant:
Hole:

64. Problems Vertical: x = -2 Horizontal : y = 1 Slant: none
Hole: at x = - 5
x int -3
y int -.5
Vertical: x = 3
Horizontal : none
Slant: y = 2x +11
Hole: none
x int: -3.5,1
y int: 7/3

65. Graph of g(x) Vertical: x = 3 Horizontal : none Slant: y = 2x +11
Hole: none int: -3.5,1
y int: 7/3

66. Find the vertical asymptotes if they exist.
Graphing a rational function.
Find the vertical asymptotes if they exist.
Find the horizontal asymptote if it exists.
Find the x intercept(s) by finding f(x) = 0.
Find the y intercept by finding f(0).
Find any crossing points on the horizontal asymptote by setting f(x) = the horizontal asymptote.
Graph the asymptotes, intercepts, and crossing points.
Find additional points in each section of the graph as needed.
Make a smooth curve in each section through known points and following asymptotes and end behavior.