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=1x+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+1x2−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=ab, 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+3x−2 the degree of both numerator and denominator are both 1, a = 3 and b = 1 and therefore the horizontal asymptote is y=31 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.