1. Graphing More Complex Rational Functions
Dr. Shildneck
Fall, 2014

2. To graph Find all vertical asymptotes Find the horizontal asymptotes
Find any slant asymptotes
Draw the asymptotes.
Plot points
Sketch Curves

3. Example 1
Graph

4. Finding Vertical Asymptotes
To find the vertical asymptotes of a rational function, find all values of x such that the denominator is zero.

5. To find the horizontal asymptote
There is at most one horizontal asymptote:
If the degree of p(x) < degree of q(x), (top power < bottom power) the horizontal asymptote is y = 0.
If the degree of p(x) = degree of q(x), (top power = bottom power) the horizontal asymptote is
If the degree of p(x) > degree of q(x), (top power > bottom power) there is NO horizontal asymptote.
Leading coefficient of p(x)
Leading coefficient of q(x)
y =

6. Example: Horizontal Asymptote
degree of top = degree of bottom = 1
2 > 1
So, there is No Horizontal Asymptote.

7. To find slant asymptotes
There are only slant asymptotes if the degree of p(x) = 1 + degree of q(x).
Divide p(x) by q(x). The result (excluding the remainder) is the equation of the slant asymptote.

8. Example: Slant Asymptotes
Degree of Top = 1 + Degree of Bottom, so
Divide:
Slant
Asymptote
– ( )
– ( )
Doesn’t matter.

9. Example: Draw Asymptotes
Vertical
Horizontal
none
Slant

10. Example: Plotting Points
Note:
There are only two sections created by vertical asymptotes. Plot points on each side of the vertical asymptote. 2 1

11. Example: Plotting Points-4 -3 -2 -1 Y
-6.5 -6 A S M
-1.5

12. Example: Plotting Points

13. Example: Sketch Branches

14. Example 2
Graph:

15. Finding Vertical Asymptotes
To find the vertical asymptotes of a rational function, find all values of x such that the denominator is zero. or

16. Example: Horizontal Asymptote
degree of top = degree of bottom = 2
2 =2, so, the Horizontal Asymptote is

17. Example: Slant Asymptotes
Degree of Top = Degree of Bottom
Thus, there is no slant asymptote.

18. Example: Draw Asymptotes
Vertical
Horizontal
Slant
none

19. Example: Plotting Points
There are three sections created by vertical asymptotes. Plot points in each section created by the vertical asymptotes.
In middle sections plot 3-5 points. 2 3 1

20. Example: Plotting Points-5 -4 -3 -2 2 3 4 5 Y
-75 16 ≈
-4.7
-48 7
-6.8 A S 12
2.4

21. Example: Plotting Points

22. Example: Sketch Branches

23. Assignment
Worksheet #4