The general equation is the graph of a parabola: y-k=a(x-h)² where (h,k) is the vertex and the focus is (h,k+1/4a)=(4,-3), and directrix is y=k-1/4a=-6. So k+1/4a=-3 and k-1/4a=-6, 2k=-9, k=-9/2. (The vertex is midway between the focus and the directrix line.)
1/4a=-3+9/2=3/2, a=1/6. So y+9/2=(1/6)(x-4)².
y=(1/6)(x²-8x+16)-9/2=x²/6-4x/3+8/3-9/2=x²/6-4x/3-11/6.
