A. Find a possible equation for the directeix.
B. Using your results from A. Find the vertex.
C. Write an equation for the parabola
This question is giving me some major difficulty I have spent 3 days on this question and I cannot solve it, and I am hoping for some assistance.
Question A:
By the definition of a parabola, the distance from the focus to any point on the parabola equals the distance from that point to the directrix.
The distance from the focus to the point $(7,-7)$ is $13$. You know the directrix is horizontal, so it has the equation $y=c$ for some constant $c$.
Which value of $c$ will make the distance from the point $(7,-7)$ to the line $y=c$ equal to $13$? Think about it: it is very easy. There are two answers to this, and it is obvious which one is the desired directrix.
Question B:
The vertex is the midpoint of the line segment from the focus to the directrix and perpendicular to the directrix. Which point is that?
Question C:
There are several ways to answer that. The easiest depends on the information you already know about the parabola. Do you know that the equation for a vertical parabola with vertex at point $(h,k)$ is
$$y-k=\frac{(x-h)^2}{4p}$$
where $p$ is the distance between the focus and the vertex?