Question

For the function, the vertex of the function's graph is given. Find the unknown coefficients.

Answer 1

$\bf \textit{vertex of a vertical parabola, using coefficients} \\\\ y=\stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{+b}x\stackrel{\stackrel{c}{\downarrow }}{+c} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)~~=~~(-2,4) \\\\[-0.35em] ~\dotfill\\\\ -\cfrac{b}{2a}=-2\implies -\cfrac{b}{2(1)}=-2\implies -b=-4\implies \boxed{b=4} \\\\[-0.35em] ~\dotfill$

$\bf c-\cfrac{ b^2}{4 a}=4\implies c-\cfrac{\boxed{4}^2}{4(1)}=4\implies c-\cfrac{16}{4}=4 \\\\\\ c-4=4\implies \boxed{c=8} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill y=x^2+4x+8~\hfill$