SOLUTION
The equation to use is:
[tex]x=\frac{1}{4 \left(f - h\right)}\left(y-k\right)^2+h[/tex]
From the given infor, f=-2, k=0
The equation becomes:
[tex]\begin{gathered} x=\frac{1}{4(-2-h)}(y-0)^2+h \\ x=\frac{1}{4(-2-h)}y^2+h \end{gathered}[/tex]
Note that the distance from the focus to vertex is equal to distance from vertex to directrix:
[tex]\begin{gathered} f-h=h-2 \\ -2-h=h-2 \\ -2+2=2h \\ h=0 \end{gathered}[/tex]
The equation becomes:
[tex]\begin{gathered} x=\frac{1}{4(-2-0)}y^2+0 \\ x=\frac{1}{-8}y^2 \\ x=-\frac{1}{8}y^2 \end{gathered}[/tex]
Therefore the required equation is:
[tex]x=-\frac{1}{8}y^2[/tex]
