Explanation:
let's see.
first, let's deal with the asymptotes.
the vertical asymptote at x = -2.
it means that for x getting close to that value y is getting enormously larger and larger, so that for x = -2 we would get what we call y = infinity. we could even accept +infinity or -infinity. in both cases we get the same asymptote
I personally think using both +infinity and -infinity is easier to define. because coming from the left (negative direction) and from the right (positive direction) we can use the same expression and don't need to worry about the sign.
so, I go for something like
y = 1/(x + 2)
or even better considering we need a horizontal asymptote too :
y = x/(x + 2)
this will divide by 0 for x = -2 creating the asymptote.
now for the horizontal asymptote of y = 3.
it means that for x getting close to infinity (+infinity and/or -infinity) y has to get closer and closer to this finite value (3).
we achieve this by dividing a dimension of x by the same dimension of x.
so, for large values of x any additive parts of the terms become irrelevant for the limit.
as we had above
y = x/(x + 2)
the limit for x going to +infinity AND to -infinity would be
x/x = 1.
but we need the limit of 3.
so, all we need to do is to multiply the expression by 3 :
y = 3x/(x + 2)
now the limit for x going to +/- infinity is
3x/x = 3
and now for the y-intersect at (0, -4) :
we need to do something, so that y = -4 for x = 0.
let's look at what we have so far :
y = 3x/(x + 2)
-4 = 3×0/(0 + 2) = 0/2 = 0
clearly, that is wrong and therefore still not covered.
imagine, we keep the denominator of the fraction unchanged.
what do we need to do on the numerator side to create -4 as result ?
in other words, we need to find an n so that
-4 = (3x + n)/(x + 2)
for x = 0
-4 = n/2
n = -8
and we get as final result
y = (3x - 8)/(x + 2)
that includes the point (0, -4), and has the asymptotes at x = -2 and y = 3.