Question

Which is the equation of the parabola that has a vertex at the origin and a focus at (3, 0)?

Answer 1

Check the picture below, so the parabola looks more or less like that, with a positive "p" distance of 3, so

`\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{`

`\begin{cases} h=0\\ k=0\\ p=3 \end{cases}\implies 4(3)(x-0)=(y-0)^2\implies 12x=y^2\implies x=\cfrac{1}{12}y^2`