If Vertex is (4,-7) and x intercept is 4, what is the standard equation of the parabola?

Answer 1

no calculus need for this. Remember your Algebra I
Vertex is (4,-7) ... y = a(x-4)^2 - 7
But the x-intercept cannot be 4, since we know that y(4) = -7

However, if you meant that the y-intercept is 4, then
a(0-4)^2 - 7 = 7
a = 7/8
and thus
y = 7/8 (x-4)^2 - 7
Now just rearrange that into standard form.