Graphing Rational Functions

A function of the form:

were p(x) and q(x) are polynomial functions; this is a rational function.

examples of rational functions:

Some suggestions on how to go about graphing rational functions:

Check for symmetry; about the y-axis or the origin or both.
Find any vertical asymptotes.
Find any horizontal asymptotes.
Find out what the graph does near the asymptotes.
Plot points to see the shape of the graph.

Finding the Vertical and Horizontal Asymptotes:

Finding the Vertical Asymptotes:
Take the denominator, make it equal to zero and solve.

ex.

The denominator is x + 1, and the vertical asymptote is:

x + 1 = 0
x = -1

The vertical asymptote is the line x = -1

Finding the Horizontal Asymptote:

From the example above.

The vertical asymptote is x = -1.

To find the horizontal asymptote, divide the variable in the numerator with the numerator and the denominator.

 

As x gets very large from negative infinity ( - ¥ ), f(x) rises slightly from 2, and as x gets smaller from positive infinity, f(x) is almost 2, and is slowly falling.

ex.

the graph is symmetric about the y-axis.

vertical asymptotes:

x²- 25 = 0
x² = 25
x = ±5

Since every value of x is squared the horizontal axis is the x-axis.