Rational functions are functions of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions. There are 7 steps to sketching a rational function: 1) find the y-intercept, 2) factor, 3) find roots of the numerator, 4) find vertical asymptotes from roots of the denominator, 5) find horizontal asymptotes, 6) determine signs over intervals, 7) sketch the graph. The appearance of the graph depends on whether the degree of the numerator is less than, equal to, or greater than the degree of the denominator.

1. Lines
Approaching
Infinity
graphing rational functions
asintoto [asymptote] by flickr user zinov

2. Graphing Rational Functions
Functions of the form
where a(x) and b(x) are polynomial functions.
Examples

3. Graphing Rational Functions
Appearance
Where n is odd, the
Where n is even, the
graph looks like this:
graph looks like this:

4. Graphing Rational Functions
Sketching (7 steps)
Step 1: Find the y-intercept (let x = 0)
Step 2: Factor everything. (Use rational roots theorem if necessary.)
Step 3: Find the roots of the function by finding the roots of the
numerator a(x).
Step 4: Find the vertical asymptotes by finding the roots of the
denominator b(x).

5. Graphing Rational Functions
Step 5: Find the horizontal asymptotes by dividing each term in the
function by the highest power of x, and take the limit as x goes to infinity.
(Use the UNfactored form.)
You will find that, in general, there are three possible results:
i When [degree of numerator < degree of denominator]
the horizontal asymptote is y = 0.
ii When [degree of numerator = degree of denominator]
the H.A. is the ratio leading coefficient of a(x)
leading coefficient of b(x)
iii When [degree of numerator > degree of denominator] there is
no horizontal asymptote; however there may be a slant asymptote or
a hole in the graph.

6. Graphing Rational Functions
Sketching (7 steps)
Step 6: Determine the sign of the function over the intervals defined by
the roots and vertical asymptotes. (Use the factored form.)
Step 7: Sketch the graph.

7. Graphing Rational Functions
Sketching: Example 1 of 4
Step 5:
Step 1:
Step 6:
Step 2:
Step 3:
Step 7:
Step 4:

8. Step 5:

9. Sketch the graph of

11. Graphing Rational Functions
Sketching: Example 2 of 4
Step 5:
Step 1:
Step 6:
Step 2:
Step 7:
Step 3:
Step 4:

12. Step 5: