Question

Find the standard form of the equation of the parabola with the given characteristics.

Vertex: (-5, 4); directrix: x = -12

Answer 1

Check the picture below, so the parabola looks more or less like so, with a "p" distance of positive 7.

`\textit{horizontal parabola vertex form with focus point distance} \\\\ 4p(x- h)=(y- k)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h+p,k)}\qquad \stackrel{directrix}{x=h-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{p~is~negative}{op ens~\supset}\qquad \stackrel{p~is~positive}{op ens~\subset} \end{cases} \\\\[-0.35em] ~\dotfill`

`\begin{cases} h=-5\\ k=4\\ p=7 \end{cases}\implies 4(7)(~~x-(-5)~~) = (~~y-4~~)^2 \implies 28(x+5)=(y-4)^2 \\\\\\ x+5=\cfrac{1}{28}(y-4)^2\implies \boxed{x=\cfrac{1}{28}(y-4)^2 - 5}`