Question

What is the equation based on the focus and vertex

Answer 1

Check the picture below, so the parabola looks more or less like so.

keeping in mind the vertex is always half-way between the focus point and the directrix, let's also notice the "p" distance.

`\textit{vertical parabola vertex form with focus point distance} \\\\ 4p(y- k)=(x- h)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h,k+p)}\qquad \stackrel{directrix}{y=k-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{`

`\begin{cases} h=-4\\ k=2\\ p=-1 \end{cases}\implies 4(-1)(y-2)=( ~~ x-(-4) ~~ )^2\implies -4(y-2)=(x+4)^2 \\\\\\ y-2=-\cfrac{1}{4}(x+4)^2\implies {\Large \begin{array}{llll} y=-\cfrac{1}{4}(x+4)^2 + 2 \end{array}}`