How to find the equation for a parabola for which you are given two points and the vertex? I was originally given the value $(4,-2)$ as the vertex of a parabola and told that it also includes the value $(3,-5)$. From this point, I deduced that the next point would have the same y-value as the point whose x-value is equidistant from the vertex, so the next point would be $(5,-5)$. I also know that since the parabola's vertex is higher than the two values surrounding it, it is a negative parabola. I am now stuck and do not know how to continue about finding the equation for this limited input/output table.

I see @Rod gave an answer using the fact that the vertex was given. I'd like to give another method that can be used to find the parabola given any three arbitrary points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$. 
Assume $y = ax^2 + bx +c$ and generate a system of three equations. I'll put it in matrix form for brevity 
$$
\left(\begin{array}{ccc}
        x_1^2 & x_1 & 1\\
        x_2^2 & x_2 & 1\\
        x_3^2 & x_3 & 1\\
\end{array}\right)
%
\left(\begin{array}{c}
         a\\
         b\\
         c\\
\end{array}\right)
%
=
\left(\begin{array}{c}
         y_1\\
         y_2\\
         y_3\\
\end{array}\right)
$$
To solve for the coefficients, just multiply by the inverse of the square matrix. In this problem we get
$$
\left(\begin{array}{c}
         a\\
         b\\
         c\\
\end{array}\right)
%
=
%
\left(\begin{array}{ccc}
        16 & 4 & 1\\
        9 & 3 & 1\\
        25 & 5 & 1\\
\end{array}\right)^{-1}
%
\left(\begin{array}{c}
         -2\\
         -5\\
         -5\\
\end{array}\right)
%
=
%
\left(\begin{array}{c}
         -3\\
         24\\
         -50\\
\end{array}\right)
$$
So therefor $y= -3x^2 + 24x - 50$ represents the equation for our parabola. 
This clearly isn't the most effective method in this case, but perhaps it would be helpful for similar problems with arbitrarily chosen points.
