Graphing rational functions can be quite challenging. One of the most important aspects of graphing rational functions is determining the asymptotes of the graph. An asymptote, generally, is a line that the graph of the function can not cross. This is always the case with vertical asymptotes, but not always the case with horizontal and oblique (slant) asymptotes.

When the degree of the numerator of a rational function is one greater than the degree of the denominator, a slant asymptote exists. In the graph you see below the oblique asymptote was found via a long division. The slant asymptote is the quotient obtained from the long division. You can see from the graph that as x gets very large in either the positive or negative direction, the graph of the rational function gets closer and closer to the slant asymptote y=x+6 or the green line. Also a vertical asymptote of x=2 exists because the rational function is undefined at x=2 , i.e. division by zero occurs when x=2.

If you want to prove or find out where a rational function may cross its oblique or slant asymptote, just set the rational function equal to the slant asymptote. If the resulting equation has solutions, then the slant asymptote will intersect the graph of the rational function at the points which correspond to the solutions. The solutions will be the x coordinates of the points of intersection.