Question

Find the equation, (f(x) = a(x-h)2+ k), for a parabola containing point (-1,0) and having (-3, 4) as a vertex. What is the standard form of the equation?

Answer 1

$\bf ~~~~~~\textit{parabola vertex form} \\\\ \begin{array}{llll} \boxed{y=a(x- h)^2+ k}\\\\ x=a(y- k)^2+ h \end{array} \qquad\qquad vertex~~(\stackrel{}{ h},\stackrel{}{ k})\\\\ -------------------------------\\\\ vertex~(-3,4)\quad \begin{cases} x=-3\\ y=4 \end{cases}\implies \stackrel{f(x)}{y}=a[x-(-3)]^2+4 \\\\\\ y=a(x+3)^2+4 \\\\\\ \textit{we also know that }(-1,0)\quad \begin{cases} x=-1\\ y=0 \end{cases}\implies 0=a(-1+3)^2+4 \\\\\\ -4=4a\implies -1=a\qquad therefore\qquad y=-(x+3)^2+4$