SOLUTION: Explain this like I'm five years old... I've tried so many pages and still don't get it. How can I find the equation of a parabola when i'm given the focus and directrix of it?

We first draw the the focus and the directrix (which is y=0 which is the x-axis,
which I have made bluish-purple below):



Next we know that since every point on a parabola is the same distance
from the vertex as it is directly to the directrix, and since the vertex
is on the parabola, it must also be exactly half way between the the
focus (0,8) and the directrix (the x-axis).  So the vertex is (0,4),
since it's halfway between the focus (0,8) and the directrix (the bluish
purple x-axis).  So we draw the vertex (0,4):



There are two other points we can plot that are exactly the same distance 
from the focus (0,4) and they are to the directrix (the x-axis).  They
are the two points (8,8) and (-8,8).  (Sometimes they are called "the ends
of the 'latus rectum', or 'focal chord', the line connecting them which I
won't bother to draw ). We plot those two points also:



Now we can sketch in the parabola since it must go through the vertex
(0,4) and those two points (8,8) and (-8,8):



The standard equation of a parabola opening upward or downward is:

 

where the vertex is (h,k). Some books use "a" instead of "p". I will 
use p. "p" (or "a") is the distance from the vertex to the focus which
is also the distance from the vertex to the directrix. p is taken to 
be positive if the parabola opens upward and negative if it opens 
downward. This one opens upward so we take p as positive. By counting 
the units we see that the vertex (0,4) is 4 units from the focus (0,8) 
and also 4 units from the directrix (the x-axis).  So p=+4, and 
(h,k) = (0,4).  So the standard equation:

 becomes upon substituting for h,k and p:

 or

 

Edwin