An equation for a rational function with the given characteristics

Question

Write an equation for a rational function with the given characteristics.

Question: vertical asymptotes at x = −3 and x = 6, x-intercepts at (−2, 0) and (1, 0), horizontal asymptote at y = −2

Answer 1

Since there are vertical asymptotes at X = -3 and X = 6, the denominator will have the terms (x+3) and (x-6)

Since the x intercepts are -2 and 1, the numerator will have the terms (x+2) and (x-1)

So far we have f(X) = a(x+2)(x-1)/(x+3)(x-6)

To find the value of A, we look at the horizontal asymptote. The horizontal asymptote describes what the function looks like when x approaches infinity, therefore a = -2 so that the limit of the function as x -> infinity will be -2

So the final answer is f(x). = -2(x+2)(x-1)/(x+3)(x-6)