12. Graphs of rational functions

A rational function is one that is defined as a ratio of polynomials, that is:

        P(x)
f(x) = ------
        Q(x)

The functions P(x) and Q(x) are polynomials. The domain of f(x) is all real numbers such that Q(x) ≠ 0. The codomain of f(x) is also the set of real numbers. The range depends on the definitions of both P(x) and Q(x). For example:

f(x) = 2x/(x² - 4)

The domain of f(x) is the set of real numbers such that x ≠ -2 and x ≠ 2. The codomain and range of f(x) is the set of real numbers.

If the polynomial Q(x) has no real zeroes, then the graph of P(x)/Q(x) is a continuous curve for all real values of x. However, if Q(x) has one or more real zeroes, the graph will have discontinuities at each real zero. Often, these discontinuities can be represented as vertical asymptotes.

Asymptotes

An asymptote is a straight line on a graph that is a limit to a curve in the sense that the distance from a moving point on the curve to the line approaches zero as the point moves an infinite distance from the origin.

Drawing graphs

Some suggestions on how to go about graphing rational functions:

Reduce the rational function to lowest terms and check for singularities, values of x for which the function is undefined.
If any factor in the denominator is identical to a factor in the numerator, then there will not be a vertical asymptote through the value for which the factor would equal zero. The function is undefined at the value. The corresponding point on the graph is usually drawn as an open circle.
Find all of the asymptotes and draw them.
How to do this is described below for vertical, horizontal, and oblique asymptotes.
Determine the symmetry of the graph.
Refer to earlier chapters if necessary.
Find and plot any intercepts on the x-axis and y-axis.
Refer to earlier chapters if necessary.
Find and plot several other points on the graph.
That is, construct a table of ordered pairs of point using {x, f(x)} and plot their locations. It is most helpful to find points on opposite sides of vertical asymptotes.
Sketch the graph one or more curves through the points found above so that the curved lines approach the asymptotes going away from the origin.
A graph may cross over a horizontal or oblique asymptote, but should not cross over a vertical asymptote.

12.1. Vertical asymptotes
12.2. Horizontal asymptotes
12.3. Oblique asymptotes
12.4. Solved problems