1. Graphing Rational Functions
3.7 Notes
Graphing Rational Functions

2. 3.7 Notes
Unlike polynomial functions which are continuous, rational functions have discontinuities.
types of discontinuities:
jump – associated with piece-wise functions
point
infinite

3. 3.7 Notes
Holes in a graph are point discontinuities. A hole is the “absence of a point” in a line or curve.

4. 3.7 Notes
Asymptotes are infinite discontinuities. Rational functions may have vertical, horizontal, and/or slant asymptotes.

5. 3.7 Notes Find and plot x and y intercepts.
Use limit theorems to find and graph the discontinuities.
a. Check for holes.
b. Check for horizontal asymptotes.
c. Check for slant asymptotes.
d. Check for vertical asymptotes.
Use limits to determine the behavior of the graph between discontinuities.
Sketch a smooth curve.

6. Finding the discontinuities:
a. Check for holes. The function may have a hole if there is a common factor in the numerator and denominator. If so, apply the theorem to find the coordinates of the hole:
If is a common factor of the numerator and denominator of f(x), then
is a hole.

7. Finding the discontinuities
b. Check for horizontal asymptotes. The function will have a horizontal asymptote if the degree of the numerator is less than or equal to the degree of the denominator. If so, apply the theorem to find the equation of the horizontal asymptote:
is a horizontal asymptote of f(x) if
or if

8. Finding the discontinuities
c. Check for slant asymptotes. The function may have a slant asymptote if the degree of the numerator is one more than the degree of the denominator. If so, apply the theorem to find the equation of the slant asymptote:
The oblique line is a slant asymptote of f(x) if
or if
when f(x) is in quotient form.

9. Finding the discontinuities
d. Check for vertical asymptotes. The function may have vertical asymptotes if the denominator is zero for some value(s) of x. If so, apply the theorem to find the equation of the vertical asymptote(s):
is a vertical asymptote of f(x) if
or if
from the left or the right.

10. 3.7 Notes Example #1: Find the discontinuities of Check for holes:
is a hole.

11. 3.7 Notes Example #1: Find the discontinuities of
Check for horizontal asymptotes:
The degree of the numerator is greater than the degree of the denominator; this rational function has no horizontal asymptotes.

12. 3.7 Notes Example #1: Find the discontinuities of
Check for slant asymptotes:
The degree of the numerator is one more than the degree of the denominator. This rational function may have a slant asymptote.

13. 3.7 Notes Example #1: Find the discontinuities of
Check for slant asymptotes:
Divide to put into quotient form:

14. 3.7 Notes Check for slant asymptotes:
Take the limit as x approaches infinity:
may be a slant asymptote. (It’s not, it is actually the graph of the function.)

15. 3.7 Notes Example #1: Find the discontinuities of
Check for vertical asymptotes:
If x = 2, the denominator is zero.

16. 3.7 Notes Check for vertical asymptotes:
x = 2 is not a vertical asymptote.
This function does not have any vertical asymptotes.

17. 3.7 Notes Example #1: Find the discontinuities of
This function has a hole at It does not have any horizontal or vertical asymptotes. It may have a slant asymptote at y = x + 5, but it doesn’t.

18. 3.7 Notes Example #2: Find the discontinuities of Check for holes:
There are no common factors in the numerator and denominator; this function has no holes.

19. 3.7 Notes Example #2: Find the discontinuities of
Check for horizontal asymptotes:
The degree of the numerator is less than the degree of the denominator; the function has a horizontal asymptote.

20. 3.7 Notes y = 0 is a horizontal asymptote.
Check for horizontal asymptotes:
y = 0 is a horizontal asymptote.

21. 3.7 Notes Example #2: Find the discontinuities of
Check for slant asymptotes:
The degree of the numerator is less than the degree of the denominator; the function has no slant asymptote.

22. 3.7 Notes Example #2: Find the discontinuities of
Check for vertical asymptotes:
If x = 0 or x = 4, the denominator is zero.

23. 3.7 Notes Check for vertical asymptotes:
x = 0 is a vertical asymptote.

24. 3.7 Notes Check for vertical asymptotes:
x = 4 is a vertical asymptote.

25. 3.7 Notes Example #2: Find the discontinuities of
This function does not have any holes. It has a horizontal asymptote whose equation is y = 0, no slant asymptote, and two vertical asymptotes whose equations are x = 0 and x = 4.

26. Practice:
Find the discontinuities of the following rational functions. 1. 2.

27. 3.7 Notes Practice #1: Find the discontinuities of Check for holes:
is a hole.

28. 3.7 Notes Practice #1: Find the discontinuities of
Check for horizontal asymptotes:
The degree of the numerator is equal to the degree of the denominator; the function has a horizontal asymptote.

29. 3.7 Notes y = 1 is a horizontal asymptote.
Check for horizontal asymptotes:
y = 1 is a horizontal asymptote.

30. 3.7 Notes Practice #1: Find the discontinuities of
Check for slant asymptotes:
The degree of the numerator is equal to the degree of the denominator; the function has no slant asymptote.

31. 3.7 Notes Practice #1: Find the discontinuities of
Check for vertical asymptotes:
If x = -3 or x = 4, the denominator is zero.

32. 3.7 Notes Check for vertical asymptotes:
x = 4 is not a vertical asymptote.

33. 3.7 Notes Check for vertical asymptotes:
x = -3 is a vertical asymptote.

34. 3.7 Notes Practice #1: Find the discontinuities of
This function has a hole at
It has a horizontal asymptote, y = 1.
It does not have a slant asymptote.
It has a vertical asymptote, x = -3.

35. 3.7 Notes Practice #2: Find the discontinuities of Check for holes:
There are no common factors in the numerator and denominator; this function has no holes.

36. 3.7 Notes Practice #2: Find the discontinuities of
Check for horizontal asymptotes:
The degree of the numerator is greater than the degree of the denominator; this rational function has no horizontal asymptotes.

37. 3.7 Notes Practice #2: Find the discontinuities of
Check for slant asymptotes:
The degree of the numerator is one more than the degree of the denominator. This rational function may have a slant asymptote.

38. 3.7 Notes Practice #2: Find the discontinuities of
Check for slant asymptotes:
Divide to put into quotient form:

39. 3.7 Notes Check for slant asymptotes:
Take the limit as x approaches infinity:
may be a slant asymptote.

40. 3.7 Notes Practice #2: Find the discontinuities of
Check for vertical asymptotes:
If x = 0 the denominator is zero.

41. 3.7 Notes Check for vertical asymptotes:
x = 0 is a vertical asymptote.

42. 3.7 Notes Practice #2: Find the discontinuities of
This function does not have a hole or a horizontal asymptote.
It has a slant asymptote, y = x – 2.
It has a vertical asymptote, x = 0.