How to Graph a Rational Function with Denominator Having the Higher Degree
After you calculate all the asymptotes and the x- and y-intercepts for a rational function, you have all the information you need to start graphing the function. In any rational function where the denominator has a greater degree, as values of x get infinitely large, the fraction gets infinitely smaller until it approaches zero (this process is called a limit).
Rational functions are really just fractions. If you look at several fractions where the numerator stays the same but the denominator gets bigger, the whole fraction gets smaller. For instance, look at 1/2, 1/20, 1/200, and 1/2,000.
When the denominator has the greater degree, you begin by graphing the information that you know for f(x):
The figure shows all the parts of the graph:
Draw the vertical asymptote(s).
Whenever you graph asymptotes, be sure to use dotted lines, not solid lines, because the asymptotes aren’t part of the rational function.