Create a rational function such that the graph of has vertical asymptotes at x=5 and x= -7, a hole at x=2 , and a horizontal asymptote at y = 14. By creating a rational function, you are to write rule for this function. There are many correct solutions here.

Answer 1

You know that the denominator has to be zero at x=5 and x=-7

The hole means that the numerator and denominator are both zero at x=2

so, we can start with

f(x) = (x-2)/[(x-2)(x-5)(x+7)]

Now, we need a horizontal asymptote at y=14. That means that the numerator and denominator must have the same degree, x^n, with the highest degree having a coefficient in the numerator 14 times that in the denominator. So, the simplest one I can think of is

f(x) = 14x^2(x-2)/[(x-2)(x-5)(x+7)]