How do you find the asymptotes of a rational function?
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
*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
*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.
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.