Question

Write the equation in vertex form the parabola with vertex (0,2) and focus (0,9)

Answer 1

Check the picture below, so the parabola looks more or less like so, with a "p" distance of 7 units, so hmmm

`\textit{vertical parabola vertex form with focus point distance} \\\\ 4p(y- k)=(x- h)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h,k+p)}\qquad \stackrel{directrix}{y=k-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{`

`\begin{cases} h=0\\ k=2\\ p=7 \end{cases}\hspace{5em}4(7)(y-2)=(x-0)^2\implies 28(y-2)=x^2 \\\\\\ y-2=\cfrac{1}{28}x^2\implies {\Large \begin{array}{llll} y=\cfrac{1}{28}x^2 + 2 \end{array}}`