Question

Write an equation for a rational function with: Vertical asymptotes at x = 6 and x = -6 x intercepts at x = 4 and x = -2 Horizontal asymptote at y = 10

Answer 1

Answer:

$f(x) = \frac{10(x-4)(x+2)}{(x-6)(x+6)}$

Step-by-step explanation:

Since it has vertical asymptotes at x = 6 and x = -6, then the dividing polynomial should have (x-6) and (x-(-6)) = (x+6) as factors. Also, the function has 2 zeros in x = 4 and x = -2, which are zeros of the numerator. As a consequence, (x-4) and (x-(-2)) = (x+2) are factors of the numerator.

If we take limit with x going to infty to the rational function

$\frac{(x-4)(x+2)}{(x-6)(x+6)}$

the result will be 1. In order for it to be 10, and therefore letting the rational function have a Horizontal asymptote in y = 10, we need to multiply that expression by 10. A possible formula for the rational function is

$f(x) = \frac{10(x-4)(x+2)}{(x-6)(x+6)}$