Let's piece this together.
y = -------------------------------
The vertical asymptotes come from zeros of the denominator (where they are not also zeros of the numerator).
We can turn zeros into factors. If c is a zero, then x - c is a factor. This gives us a denominator of
y = -------------------------------
(x - 4)(x + 6)
The x-intercepts come from the zeros of the numerator (where they are not also zeros of the denominator). We can turn these zeros into factors. This gives us
(x + 2)(x + 5)
y = -------------------------------
(x - 4)(x + 6)
Lastly, we need a y-intercept at 2. If we substitute zero for x we can find the y-intercept currently. This amounts to multiplying all the constants. We get
(2)(5) 10 -5
-------------- = ----- = -----
(-4)(6) -24 12
We need a constant factor, a, in order to get the y-intercept to be 2. We solve
(-5/12)a = 2
Multiplying both sides by the reciprocal of (-5/12) we get
a = 2(-12/5) = -24/5
We can put a factor of -24 in the numerator and a factor of 5 in the denominator.
-24(x + 2)(x + 5)
y = -------------------------------
5(x - 4)(x + 6)
If we multiply the constants in the numerator we get (-24)(2)(5) = -240
If we multiply the constants in the denominator we get (5)(-4)(6) = -120
(-240)/(-120) = 2, therefore we have the correct y-intercept. We are finished!
