Question

How does the a, h and k variables affect a parabola?

Answer 1

and how do the same variables affect a quadratic graph?

Answer 2

assuming you mean a parabola of the form

y = a(x-h)^2+k

Then the vertex is at (h,k).
The graph is that of y=x^2, but vertically scaled by a factor of a, and shifted right by h and up by k.

As for general quadratics, the question makes no sense.