A comprehensive graph of a rational function will
exhibits these features:
all intercepts, both x and y;
location of all asymptotes: vertical, horizontal, and/or oblique;
the point at which the graph intersects its non-vertical asymptote (if there is such a point);
enough of the graph to exhibit the correct end behavior (i.e. behavior as the graph approaches its nonvertical asymptote). To find asymptotes of a rational function defined by a rational
expression in lowest terms, use the following procedures:
Vertical Asymptotes
Set the denominator equal to 0 and solve for x. If a is a zero of the denominator but not the numerator, then the line x = a is a vertical asymptote.
Other Asymptotes Consider three possibilities:
If the numerator has lesser degree than the denominator, there is a horizontal asymptote, y = 0 ( the x-axis).