In order to find the vertical asymptotes of a rational function, you need to have the function in factored form. You also will need to find the zeros of the function. For example, the factored function $y=\frac{x+2}{(x+3)(x-4)}$ has zeros at x = – 2, x = – 3 and x = 4.

If the numerator and denominator have no common zeros, then the graph has a vertical asymptote at each zero of the denominator. In the example above $y=\frac{x+2}{(x+3)(x-4)}$, the numerator and denominator do not have common zeros so the graph has vertical asymptotes at x = – 3 and x = 4.

If the numerator and denominator have a common zero, then there is a hole in the graph or a vertical asymptote at that common zero.
Examples:
1. $y=\frac{(x+2)(x-4)}{x+2}$ is the same graph as y = x – 4, except it has a hole at x = – 2.

2.$y=\frac{(x+2)(x-4)}{(x+2)(x+2)(x-4)}$ is the same as the graph of $y=\frac{1}{x+2},$except it has a hole at x = 4. The vertical asymptote is x = – 2.

To Find Horizontal Asymptotes:

• The graph has a horizontal asymptote at y = 0 if the degree of the denominator is greater than the degree of the numerator. Example: In $y=\frac{x+1}{x^2-x-12}$ (also $y=\frac{x+1}{(x+3)(x-4)}$ ) the numerator has a degree of 1, denominator has a degree of 2. Since the degree of the denominator is greater, the horizontal asymptote is at $y=0$.
• If the degree of the numerator and the denominator are equal, then the graph has a horizontal asymptote at $y=\frac{a}{b}$, where a is the coefficient of the term of highest degree in the numerator and b is the coefficient of the term of highest degree in the denominator. Example: In $y=\frac{3x+3}{x-2}$ the degree of both numerator and denominator are both 1, a = 3 and b = 1 and therefore the horizontal asymptote is $y=\frac{3}{1}$ which is $y=3$
• If the degree of the numerator is greater than the degree of the denominator, then the graph has no horizontal asymptote.