Vertical asymptotes are caused when factors in the denominator of a rational function are zero. For this example:
| vertical asymptote | x = 1 | x = 3 |
| factor | (x - 1) = 0 | (x - 3) = 0 |
An x-intercept occurs when the when factors in the numerator are zero. So:
| x-intercept | x = 2 | x = 5 |
| factor | (x - 2) = 0 | (x - 5) = 0 |
A horizontal asymptote exists for any function where the degree of the largest term in the numerator is less than or equal to the largest degree in the denominator. If the degree in the numerator is less than the degree in the denominator then the asymptote is at y = 0. If the degree of the numerator is equal to the degree of the denominator, then the asymptote exist at the ratio of the coefficient of the term with the largest degree in the numerator to the coefficient of the term with the largest degree in the denominator. In this case:
| horizontal asymptote | y = 7 |
| ratio of coefficients | 7 |
To satisfy all of these requirements the rational function would be:
f(x) = 7(x - 2)(x - 5) / [(x - 1)(x - 3)] = (7x2 - 49x +70)/(x2 - 4x + 3)
