Question

Find a rational equation in factored form with:
- x-intercepts at x= 3 and x=-1
- y-intercept at y= 1/6
- horizontal asymptote at y=1
- Vertical asymptote at x=2 and x= -3

Answer 1

vertical asymptotes:
y = 1/ (x-2)(x-3)

x-intercepts:
y = (x-3)(x+1) / (x-2)(x-3)
that has a horizontal asymptote at y=1

Now, how about that pesky y-intercept at (0,1/6)?
How will you fix that up?

Answer 2

Oops. Excuse my typo. It should have said
y = (x-3)(x+1) / (x-2)(x+3)

Now, we need to move the y-intercept, without moving the horizontal asymptote. How can we multiply y by 1/3 when x=0, but not change y for large x? This will do the trick:

y = (x-3)(x+1) / (x-2)(x-3) * (x^2+1)/(x^2+3)

that does not move any of the other features required, but does change the y-intercept. See the graph at

https://www.wolframalpha.com/input/?i=%28%28x-3%29%28x%2B1%29%29+%2F+%28%28x-2%29%28x%2B3%29%29+*+%28x%5E2%2B1%29%2F%28x%5E2%2B3%29