This is the question:
Shown in the figure below is the graph of a rational function with vertical asymptotes x=2, x=6, and horizontal asymptote y= -2 . (All x-intercepts of the graph of f are also shown, and a point on the graph is indicated.) The equation for f(x) has one of the five forms shown below. Choose the appropriate form for f(x) , and then write the equation.
I can't get the graph on but point are at (-4,0); (0,2); (3,0).
The answers choices are:
A) f(x)= a / x-b
B) f(x)= a(x-b) / x-c
C) f(x)= a / (x-b)(x-c)
D) f(x)= a(x-b) / (x-c)(x-d)
E) f(x)= a(x-b)(x-c) / (x-d)(x-e)
The Attempt at a Solution
What I first did was try and get rid of equations that couldn't possible work. So I knew I had vertical asymptotes at x=2,6. I had two asymptotes so I knew that options A and B couldn't work. I then got rid of option C because dividing a by x to get my horizontal asymptote would not give me y= -2 it would give me y=0. I then got rid of option E by factoring the top part through. Since having ax^2 as my leading coificent would give me a diagonal asymptote I got rid of it. So I got D as my answer but then I have to find the equation. So I got (x-2)(x-6) for the bottom because those would give me vertical asymptotes at x=2,6 But I don't know how to find the top. I got -2(x-12) divided by (x-2)(x-6) because sticking 0 in for x would give me 2. So I had the point (0,2) but I can't get any of the others to match up. I'm not sure how to figure it out.