1. Rational Functions

2. Definition: A Rational Function is a function in the form:
f(x) =
where p(x) and q(x) are polynomial functions and q(x) ≠ 0.
In this section, p and q will have degree 1 or 0.
For example:

3. Very Important definitions:
Vertical asymptote occurs at values of x for which the function is undefined (exception: unless there is a hole we’ll talk about that later).
Horizontal asymptote occurs if the function approaches a specific value when x approaches infinity or negative infinity. Think of the warmup: what happens to y when x gets REALLY big or REALLY small?
To graph rational functions, ALWAYS figure out the asymptotes FIRST. Then you can plot specific points!!!!

4. Vertical asymptote will occur at x = 0.
Consider:
Vertical asymptote will occur at x = 0.
Df = (, 0), (0, )
Horizontal asymptote will occur at y = 0.
Think what happens when you divide 1 by a VERY large number!!!!!
Show your asymptotes!!
Pick some x’s on each side of the vertical asymptote to see the graph!!!

5. Note: the graph represents a hyperbola centered at (0, 0)x y Y −3 3 −2 2 −1 1
−.5 .5
-.3333
.3333
-.5 .5 -1 1 -2 2
In most cases, the range will be closely related to the horizontal asymptote be sure to check the graph.
Rf = (, 0), (0, )
Note: the graph represents a hyperbola centered at (0, 0)

6. Another type of rational function:
The vertical asymptote is still x = h.
Df = (, h), (h, )
Based on our observations, the hortizontal asymptote is y = k.
So, this will be a hyperbola centered at (h, k)!!

7. Vertical asymptote: x = –3
Horizontal asymptote: y = 2
Show your asymptotes!!
Pick some x’s on each side of the vertical asymptote to see the graph!!! x y Y −4 –2 −5 –1 4 1 3
Use more points if you want . . .
Df = (, 3), (3, )
Rf = (, 2), (2, )

8. Last example: Asymptotes: x = –2 y = 3 8 –2 5.5 .5
Vertical asymptote will correspond to the value that makes the denominator 0.
Horizontal asymptote: y =
Asymptotes:
x = –2
y = 3 x y Y −3 –1 −4 8 –2
5.5 .5
Df = (, 2), (2, )
Rf = (, 3), (3, )

9. x = 0; y = 0
x = 3; y = 1

10. x = 4; y = 1
Remember: Find the asymptotes FIRST. Show them on the graph!!!
Pick x values to the right and to the left of the vertical asymptote(s).
Use the points along with the asymptotes to sketch the graph!!!