This document discusses rational functions, which are defined as the ratio of two polynomials. It provides examples of specific rational functions and examines their key properties including vertical and horizontal asymptotes. It discusses how to predict asymptotes from the polynomials and emphasizes the importance of graphing to verify predictions. Guidelines are provided for accurately graphing rational functions on graph paper including showing intercepts, extrema, asymptotes, holes, and using proper scaling.

1. 3.6 Rational Functions
Proverbs 24:20 for the evil man has no
future; the lamp of the wicked will be put out.

2. A rational function is the ratio of two polynomials
P(x)
r(x) =
Q(x)

3. A rational function is the ratio of two polynomials
P(x)
r(x) =
Q(x)
1
Consider: y=
x

4. A rational function is the ratio of two polynomials
P(x)
r(x) =
Q(x)
1
Consider: y=
x
Graph and consider the following:
as x → ∞ , y → 0 +
as x → −∞ , y → 0
−
+
as x → 0 , y → ∞
−
as x → 0 , y → −∞

5. 1
Consider: y=
x

6. 1
Consider: y=
x
Vertical Asymptote at x = 0
Horizontal Asymptote at y = 0

7. 1
Consider: y=
x
Vertical Asymptote at x = 0
Horizontal Asymptote at y = 0
1
Consider y = with specific values ...
x

8. 1
Consider: y=
x
Vertical Asymptote at x = 0
Horizontal Asymptote at y = 0
1
Consider y = with specific values ...
x
1
y=
big #

9. 1
Consider: y=
x
Vertical Asymptote at x = 0
Horizontal Asymptote at y = 0
1
Consider y = with specific values ...
x
1
y= gets small
big #

10. 1
Consider: y=
x
Vertical Asymptote at x = 0
Horizontal Asymptote at y = 0
1
Consider y = with specific values ...
x
1 1
y= gets small y=
big # small #

11. 1
Consider: y=
x
Vertical Asymptote at x = 0
Horizontal Asymptote at y = 0
1
Consider y = with specific values ...
x
1 1
y= gets small y= gets big
big # small #

12. 1
Consider: y=
x
Vertical Asymptote at x = 0
Horizontal Asymptote at y = 0
1
Consider y = with specific values ...
x
1 1
y= gets small y= gets big
big # small #
This function is helpful in understanding
more complex rational functions.

13. Vertical Asymptotes occur when a zero is present
in the denominator.

14. Vertical Asymptotes occur when a zero is present
in the denominator.
*exception*: if the zero factor is also
present in the numerator, a hole occurs.

15. Vertical Asymptotes occur when a zero is present
in the denominator.
*exception*: if the zero factor is also
present in the numerator, a hole occurs.
Graph:
2
x + 2x − 3
y= 2
x + 5x + 6

16. Vertical Asymptotes occur when a zero is present
in the denominator.
*exception*: if the zero factor is also
present in the numerator, a hole occurs.
Graph:
2
x + 2x − 3
y= 2
x + 5x + 6

17. Vertical Asymptotes occur when a zero is present
in the denominator.
*exception*: if the zero factor is also
present in the numerator, a hole occurs.
Graph:
2
x + 2x − 3 (x + 3)(x − 1)
y= 2 y=
x + 5x + 6 (x + 3)(x + 2)

18. Vertical Asymptotes occur when a zero is present
in the denominator.
*exception*: if the zero factor is also
present in the numerator, a hole occurs.
Graph:
2
x + 2x − 3 (x + 3)(x − 1)
y= 2 y=
x + 5x + 6 (x + 3)(x + 2)
V.A. at x = −2
Hole at x = −3

19. Horizontal Asymptotes occur as follows:

20. Horizontal Asymptotes occur as follows:
a) if degree of numerator < degree of denom.
then H.A. at y = 0

21. Horizontal Asymptotes occur as follows:
a) if degree of numerator < degree of denom.
then H.A. at y = 0
b) if degree of numerator > degree of denom.
then no H.A.

22. Horizontal Asymptotes occur as follows:
a) if degree of numerator < degree of denom.
then H.A. at y = 0
b) if degree of numerator > degree of denom.
then no H.A.
c) if degree of numerator = degree of denom.
leading coeff . of num.
then H.A. at y =
leading coeff . of den.

23. Predict asymptotes, then graph to verify

24. Predict asymptotes, then graph to verify
2
3x + x + 12
1) P(x) = 2
x − 5x + 4

25. Predict asymptotes, then graph to verify
2
3x + x + 12
1) P(x) = 2
x − 5x + 4
2
3x + x + 12
P(x) =
(x − 4)(x − 1)

26. Predict asymptotes, then graph to verify
2
3x + x + 12
1) P(x) = 2
x − 5x + 4
2
3x + x + 12
P(x) =
(x − 4)(x − 1)
V.A. at x = 4, x = 1
H.A. at y = 3

27. Predict asymptotes, then graph to verify
2
2x + x − 3
2) y= 2
x + 2x − 3

28. Predict asymptotes, then graph to verify
2
2x + x − 3
2) y= 2
x + 2x − 3
(2x + 3)(x − 1)
y=
(x + 3)(x − 1)

29. Predict asymptotes, then graph to verify
2
2x + x − 3
2) y= 2
x + 2x − 3
(2x + 3)(x − 1)
y=
(x + 3)(x − 1)
V.A. at x = −3
H.A. at y = 2
Hole at x = 1

30. Predict asymptotes, then graph to verify
3x + 6
3) y= 2
x − 4x + 4

31. Predict asymptotes, then graph to verify
3x + 6
3) y= 2
x − 4x + 4
3(x + 2)
y= 2
(x − 2)

32. Predict asymptotes, then graph to verify
3x + 6
3) y= 2
x − 4x + 4
3(x + 2)
y= 2
(x − 2)
V.A. at x = 2
H.A. a y=0

33. Predict asymptotes, then graph to verify
4 3
x − x +1
4) y=
x −1

34. Predict asymptotes, then graph to verify
4 3
x − x +1
4) y=
x −1
1 1 -1 0 0 1
1 0 0 0
1 0 0 0 1 not a factor

35. Predict asymptotes, then graph to verify
4 3
x − x +1
4) y=
x −1
1 1 -1 0 0 1
1 0 0 0
1 0 0 0 1 not a factor
V.A. at x = 1

36. When doing graphs on graph paper:

37. When doing graphs on graph paper:
1) Use correct x and y intercepts

38. When doing graphs on graph paper:
1) Use correct x and y intercepts
2) Use correct local extrema

39. When doing graphs on graph paper:
1) Use correct x and y intercepts
2) Use correct local extrema
3) Show and label all vertical asymptotes

40. When doing graphs on graph paper:
1) Use correct x and y intercepts
2) Use correct local extrema
3) Show and label all vertical asymptotes
4) Show and label all horizontal asymptotes

41. When doing graphs on graph paper:
1) Use correct x and y intercepts
2) Use correct local extrema
3) Show and label all vertical asymptotes
4) Show and label all horizontal asymptotes
5) Show and label any holes

42. When doing graphs on graph paper:
1) Use correct x and y intercepts
2) Use correct local extrema
3) Show and label all vertical asymptotes
4) Show and label all horizontal asymptotes
5) Show and label any holes
6) Use dashed lines for asymptotes

43. When doing graphs on graph paper:
1) Use correct x and y intercepts
2) Use correct local extrema
3) Show and label all vertical asymptotes
4) Show and label all horizontal asymptotes
5) Show and label any holes
6) Use dashed lines for asymptotes
7) Show you x-scale and y-scale

44. HW #7
“In the middle of difficulty lies opportunity.”
Albert Einstein