Question

Determine the equation in standard form for the quadratic function shown

Answer 1

from the picture above, we can see its vertex or U-turn is at (-1 , 65) whilst it has a point at (2 , 28), so we can say

`~~~~~~\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=-1\\ k=65\\ \end{cases}\implies y=a(~~x-(-1)~~)^2 + 65\hspace{4em}\textit{we also know that} \begin{cases} x=2\\ y=28 \end{cases} \\\\\\ 28=a(~~2-(-1) ~~ )^2 + 65\implies 28=a(3)^2 + 65\implies -37=a9 \\\\\\ \cfrac{-37}{9}=a\hspace{5em}y=-\cfrac{37}{9}(~x-(-1)~)^2 + 65\implies \boxed{y=-\cfrac{37}{9}(x+1)^2 + 65}`

Check the picture below.