Question

Please find the equation for this parabola. Thank you!

Answer 1

Check the picture below.

so we know it has a vertex at (2 ,3 ) and it also passes through (-2 , -1), then

`~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{a~is~negative}{op ens~\cap}\qquad \stackrel{a~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} h=2\\ k=3\\ \end{cases}\implies y=a(~~x-2~~)^2 + 3\hspace{4em}\textit{we also know that} \begin{cases} x=-2\\ y=-1 \end{cases}`

`-1=a(-2-2)^2 + 3\implies -4=a(-4)^2\implies -4=16a \\\\\\ \cfrac{-4}{16}=a\implies -\cfrac{1}{4}=a\hspace{10em}\boxed{y=-\cfrac{1}{4}(x-2)^2 + 3}`