Question

Find the equation of a parabola that has a vertex of (-3,2) and contains the point (4,7)

Answer 1

$\bf ~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{"a"~is~negative}{op ens~\cap}\qquad \stackrel{"a"~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill$

$\bf \stackrel{vertex}{(-3~~,~~2)}~~ \begin{cases} h = -3\\ k = 2 \end{cases}\implies y=a[x-(-3)]^2+2\implies y =a(x+3)^2+2 \\\\\\ \stackrel{\textit{passes through}}{(4~~,~~7)}~~ \begin{cases} x = 4\\ y = 7 \end{cases}\implies 7 = a(4+3)^2+2\implies 7 = 49a+2 \\\\\\ 5=49a\implies \cfrac{5}{49}=a~\hfill \boxed{y = \cfrac{5}{49}(x+3)^2+2}$