How do you find the asymptotes of a rational function?

1 Answer
Sep 15, 2014

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.