Question

Suppose a parabola has a vertex (5,-3) and also passes through (6,1). Write the equation of the parabola in vertex form

Answer 1

so hmmmm let's assume is a vertical parabola, so, the squared variable will then be the "x".


`\bf \qquad \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\ ({{ h}},{{ k}})\\\\ -------------------------------\\\\ \begin{cases} h=5\\ k=-3 \end{cases}\implies y=a(x-\stackrel{h}{5})^2\stackrel{k}{-3} \\\\\\ \textit{now, we also know that } \begin{cases} x=6\\ y=1 \end{cases}\implies 6=a(1-5)^2-3 \\\\\\ 9=a(-4)^2\implies 9=16a\implies \cfrac{9}{16}=a\qquad thus\quad \boxed{y=\cfrac{9}{16}(x-5)-3}`