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.

Note that two points are not sufficient alone to determine a unique parabola (just as two points do not determine a unique circle). However, if you also assume that the parabola is "vertical", the extra condition does uniquely determine the parabola.
Assuming the parabola is "vertical", then it is the graph of an equation of the form $$y=c(x-a)^2+b$$ where the vertex is the point $(a,b)$. You are told that $a=4$ and $b=-2$, so the equation is of the form $$y=c(x-4)^2 -2$$
Plug in the other point known to lie on the parabola to find $c$: 
$$-5 =c(3-4)^2-2$$ $$-5=c-2$$ $$c=-3$$
So an equation of the parabola is $$y=-3(x-4)^2-2$$
This can be "simplified" if desired to get $$y=-3x^2 +24x-50$$
