Graphing Rational Functions
To graph a rational function, we must determine three things:
- Zeros--x values for which the numerator equals 0 (but not the
- Vertical asymptotes--x values for which the denominator equals 0 (but
not the numerator).
- Holes--x values for which the numerator and the denominator equal
Note: If a value of x makes a squared term in the denominator equal to
0, that value is called a "double asymptote." For example, f (x) = has a double asymptote of x = 4.
Here are the steps to graphing a rational function:
- Plot zeros.
- Graph vertical asymptotes. These divide the graph into "sections."
- Start from the right side of the graph. If the
degree of the numerator is greater than the
degree of the denominator, start from the upper right corner (or the lower right
corner if the function is negative). If the degree of the numerator is less
than the degree of the denominator, start just above the x-axis (or just below
if the function is negative). If the degree of the numerator is equal to the
degree of the denominator, start just above the line y = k, where k is the
leading coefficient (or just below if negative).
- Cross over any zeros, and approach the first asymptote.
- If the asymptote is a single asymptote, approach on the opposite side of the
asymptote from the opposite direction (up if the last asymptote led down, and
vice versa). If the asymptote is a double asymptote, approach from the same
- Cross over any zeros, and approach the next asymptote.
- Repeat steps 5 and 6 until the end of the graph is reached.
- Remove all holes.
Example: Graph f (x) = .
- Zeros: x = - 1, x = 0 (double), x = 5
- Asymptotes: Single: x = 4. Double: x = - 2.
- Holes: x = 3.
- Degree of numerator = 5. Degree of denominator = 4.
Steps 1 and 2
Steps 5 and 6 (section 2)
Steps 5 and 6 (section 3)