Question

How do you find the asymptotes of a rational function?

Answer 1

To Find Vertical Asymptotes:

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 = (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 = (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= ((x+2)(x-4))/(x+2)` is the same graph as y = x - 4, except it has a hole at x = - 2.

2.`y= ((x+2)(x-4))/((x+2)(x+2)(x-4))` is the same as the graph of `y = 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=(x+1)/(x^2-x-12)` (also `y=(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 = 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=(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=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.