Focus lies on the axis of symmetry so x=1 is the axis of symmetry.
The vertex lies on the axis of symmetry and is equidistant from the focus and directrix line. This makes the vertex y coord midway between -2 and 2, which is y=0. The x coord is x=1 so the vertex is (1,0).
The focal length, f, is 2 and this gives us the final piece of information: y=a(x-1)^2, a=1/4f=1/8.
The equation is 8y=(x-1)^2 or y=(1/8)(x-1)^2.