Since the directrix is a vertical line (x=7), the parabola opens horizontally.
The vertex of the parabola is the midpoint between the focus and the directrix.
The x-coordinate of the vertex is halfway between the focus's x-coordinate and the directrix's x-coordinate:
x=(3+7)/2=5
The y-coordinate of the vertex is the same as the focus's y-coordinate:
y=-3
The distance between the vertex and the focus (or the vertex and the directrix) is called the "p" value of the parabola. Since the parabola opens horizontally, p represents the distance between the vertex and the focus:
p=3
Now we can use the equation:
(x−h)^2=4p(y−k)
where (h,k) is the vertex of the parabola.
Plugging in the values we found:
(x−5)^2=4(3)(y+3)
Simplifying:
(x−5)^2=12(y+3)
This is the equation of the parabola in vertex form. It can also be written in standard form, if desired:
y=-1/12(x−5)^2−1/4