Question 1081914
From the definition of parabola, you can find the answer,
without needing to resort to a memorized formula.
The definition says that a parabola is the set of all points
whose distances to vertex and d directrix are equal.
1) The vertex must be halfway between y=-4 and focus (3,-7),
so the vertex is (3,-5.5).
That point satisfies
{{{(x-3)^2=-6(y+5.5)}}} and {{{(x-3)^2=-1.5(y+5.5)}}} ,
but not the other two equations.
2) A point with y=-7 is at a distance
-4 - (-7) = -4 + 7 = 3 from the directrix,
so it must be at distance 3 from focus (3,-7).
That only happens for points (0,-7) and (6,-7).
Substituting x=0 and y=-7, you find that
{{{(x-3)^2=(0-3)^2=9}}} equals
{{{-6(y+5.5)=-6(-7+5.5)=-6*(-1.5)=9}}} ,
so {{{(x-3)^2=-6(y+5.5)}}} is the equation you are looking for.
 
Your teacher may have told you that the equation of a parabola
with vertex (h,k), directrix y=k-a and focus (h,k+a) is 
{{{(x-h)^2=4a(y-k)}}} .
If so, and if you memorized that formula,
you easily see that in this case h=3.
You can find k=-5.5 as the average of
the -4 in y=-4, and the -7 in (3,-7),
and you know that k+a=-5.5+a
is the y-coordinate of the focus,
so -5.5+a = -7, meaning a= -7 +5.5 = -1.5.
Applying the memorized formula gives you the same result,
without needing to understand why you do what you do.