Question

Write an equation for a rational function with: Vertical asymptotes at x = 5 and x = -4 x intercepts at x = -6 and x = 4 Horizontal asymptote at y = 9?

Answer 1

Rational function is `(9x^2+18x-216)/(x^2-x-20)`

Explanation:

(1) As we have vertical asymptotes at `x=5` and `x=-4`, we have in denominator `(x-5)` and `(x+4)` as factors

(2) As we have `x`-intercepts at `x=-6` and `x=4`, we have `(x+6)` and `(x-4)` in numerator as factors

(3) As we have horizontal asymptote at `y=9`, the highest power of numerator and denominator are same and coefficient of numerator is `9` times that of denominator

Hence, rational function is `(9(x+6)(x-4))/((x-5)(x+4))` or `(9x^2+18x-216)/(x^2-x-20)`

graph{(9x^2+18x-216)/(x^2-x-20) [-41.83, 38.17, -10.08, 29.92]}