Question

write the equation of the line that passes through the vertex of the parabola y=-2x^2+12x-11 and the point (-2,47)

Answer 1

$\bf \textit{vertex of a vertical parabola, using coefficients} \\\\ y=\stackrel{\stackrel{a}{\downarrow }}{-2}x^2\stackrel{\stackrel{b}{\downarrow }}{+12}x\stackrel{\stackrel{c}{\downarrow }}{-11} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right) \\\\\\ \left(-\cfrac{12}{2(-2)}~~,~~-11-\cfrac{12^2}{4(-2)} \right)\implies \left( 3~~,~~-11+\cfrac{144}{8} \right) \\\\\\ (3~~,~~-11+18)\implies (3~~,~~7) \\\\[-0.35em] ~\dotfill$

$\bf \stackrel{vertex}{(\stackrel{x_1}{3}~,~\stackrel{y_1}{7})}\qquad (\stackrel{x_2}{-2}~,~\stackrel{y_2}{47}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{47}-\stackrel{y1}{7}}}{\underset{run} {\underset{x_2}{-2}-\underset{x_1}{3}}}\implies \cfrac{40}{-5}\implies -8$

$\bf \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{7}=\stackrel{m}{-8}(x-\stackrel{x_1}{3}) \\\\\\ y-7=-8x+24\implies y=-8x+31$