Question

The vertex of a parabola is (0, 0) and the focus is (1/8,0). What is the equation of the parabola?

Answer 1

if the vertex is at the origin, and the focus point horizontally to the right-side 1/8 units away from the vertex, well, that means is a horizontal parabola with a "p" value of +1/8, 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{p~is~negative}{op ens~\supset}\qquad \stackrel{p~is~positive}{op ens~\subset} \end{cases} \\\\[-0.35em] ~\dotfill`

`\begin{cases} h=0\\ k=0\\ p=(1)/(8) \end{cases}\implies 4((1)/(8))(~~x-0~~) = (~~y-0~~)^2\implies \cfrac{1}{2}x=y^2\implies \boxed{x=2y^2}`