Considering the function below. What transformations should occur on a graph to have the following properties?

Question

1. f(x) = 9/x^2 - 9

Horizontal Asymptote: y = 3

Vertical Asymptote: x = 5, x = -1

Answer 1

f(x) = 9/(x²-9) = 9/(x+3)(x-3)

f(x) has a horizontal asymptote at y = 0 and vertical asymptotes are x = -3 and +3.

To get a horizontal asymptote at y = 3, just add 3 to the whole equation. That'll move the horizontal asymptote to y = 0+3 = 3:

f(x) = 9/(x²-9) + 3

To get vertical asymptotes at x = 5 and -1, shift the whole function to the right by 2 by subtracting 2 from the x terms. That'll shift the x=-1 asymptote from -3 to-3+2 = -1 and the x = 3 asymptote to x = 3+2 = 5.

f(x) = 9/((x-2)²-9)) + 3

You may want to multiply the (x-2)² out and combine like terms in the denominator.