Find an equation of a parabola with the vertex at (3, -8) and (5, -2).
I assume you mean the vertex is at (3,-8) and the graph passes through (5,-2)
The vertex tells us that
y = a(x-3)^2 - 8
Now just plug in the point to solve for a:
-2 = a(5-3)^2 - 8
4a = 6
a = 3/2
y = 3/2 (x-3)^2 - 8
and the graph shows this at
http://www.wolframalpha.com/input/?i=plot+y%3D-8%2Cy%3D3%2F2+%28x-3%29^2+-+8+%2C+y+%3D+-2