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.

Vertex form: $g(x) = a(x-h)^2+k$ where $(h,k)$ is your vertex.
Here $$g(x) = a(x-4)^2-2 $$ and $$g(3) = -5 \Rightarrow -5 = a(3-4)^2-2 \Rightarrow -3 = a \Rightarrow g(x) = -3(x-4)^2-2.$$
