1. Graphing Rational Functions
Special Cases
And
Characteristics

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.
So NO vertical asymptote!

5. Example: Horizontal Asymptote
degree of top = degree of bottom = 2
So, the Horizontal Asymptote is based on leading coefficient:
y = 6

6. Example: Draw Asymptotes
Vertical
none
Horizontal
y = 6

7. Example: Plotting Points-2 -1 1 2 Y

8. Example: Plotting Points

9. Example: Sketch Curve

10. HOLES!!!
Graph:

11. Finding Asymptotes and Holes
If the numerator and denominator of a function are both factorable, you should do that FIRST!

12. Finding Asymptotes and Holes
Then you need to simplify

13. Rules for Holes and Asymptotes
If a common factor cancels top and bottom, there will be a hole at the x-value that makes that factor equal to zero.
- Find the point by plugging that value into the
reduced form of the function.
Once you simplify (out the hole), graph what’s left just like any other rational function.

14. Example: Find all Holes and Asymptotes
Hole(s) Asymptotes
The point: VA:
Hole: HA:

15. Ex: Draw Asymptotes and Holes
Vertical
Horizontal
Hole
(-3, -2/5)

16. Example: Plotting Points1 2 3 4 Y A S Y

17. Ex: Plot Points and Sketch Branches
Vertical
Horizontal
Hole
(-3, -2/5)

18. Graphing WS (with special cases included)
Assignment
Graphing WS (with special cases included)