I know how to graph these kinds of equations with parabolas but cannot figure out how to find the factored, vertex, and standard forms of these equations.

Like this?

y = -x^2 + 4x

If so,

Standard form = -(x^2) + 4x

For vertex form,

Factor out the - sign to get the reflection/stretch (this is "a" in y=a(x-h)^2 + k

y = -(x^2 - 4x)

Complete the square:

y = -(x^2 - 4x + 4) + 4 (adding NEGATIVE 4 inside to complete the square. Do not forget the - factored outside the parentheses. Since you added -4 to complete the square, you must add +4 outside to have a net effect of nothing being added or subtracted on that side of the equation.

y = -(x-2)^2 + 4

Vertex = (2,4)