Graphical Behavior

Horizontal Asymptotes

Remember a horizontal asymptote is a horizontal line that the graph of the function approaches, but unlike vertical asymptotes the graph can cross horizontal asymptotes. To determine the horizontal asymptotes of a rational function, compare the degree of the numerator to the degree of the denominator.

Rule 1 If the degree of the numerator is smaller than the degree of the denominator, then the horizontal asymptote has the equation y = 0.

Example. Determine the horizontal asymptote of the function.

Since the numerator is first degree (linear) and the denominator is second degree (quadratic), the degree of the numerator is smaller than the degree of the denominator. Therefore, the function has a horizontal asymptote at the x-axis (y = 0).

Notice that the function crosses its horizontal asymptote.