SOLUTION
From the focus (-3, 3) and the directrix y = 7 given, note that the vertex is usually between the focus and the directrix.
So, the vertex will have the same x-coordinate as the focus, which is -3, and the y-coordinate of the vertex becomes
[tex]\begin{gathered} \frac{3+7}{2} \\ that\text{ is 3 from the y-coordinate of the focus and 7 from the directrix} \\ y=7 \\ \frac{3+7}{2}=\frac{10}{2}=5 \end{gathered}[/tex]
Hence coordinate of the vertex is (-3, 5)
Now, equation of a parabola is given as
[tex]\begin{gathered} (x-h)^2=4p(y-k) \\ where\text{ \lparen h, k\rparen is the coordinate of the vertex and p is the focal length} \\ y=k-p,\text{ so we have } \\ 7=5-p \\ p=5-7=-2 \end{gathered}[/tex]
So putting in the values of h, k and p into the equation, we have
[tex]\begin{gathered} (x-h)^{2}=4p(y-k) \\ (x-(-3)^2=4(-2)(y-5) \\ (x+3)^2=-8(y-5) \\ -\frac{1}{8}(x+3)^2=y-5\text{ that is dividing through by -8} \\ making\text{ y the subject, we have } \\ y=-\frac{1}{8}(x+3)^2+5 \end{gathered}[/tex]
Hence the answer is
[tex]y=-\frac{1}{8}(x+3)^{2}+5[/tex]
