Question

Find the standard form of the equation of the parabola with a vertex at the origin and a focus at (0, -7).

Answer 1

So hmm check the picture below

we know the vertex is at the origin, and the focus point is below it, that means two things, the parabola is vertical and it's opening downwards

notice the distance "p", from the vertex to the focus point, is just 7 units, however, since the parabola is opening downwards, the "p" value will be negative, so p = -7

$\bf \textit{parabola vertex form with focus point distance}\\\\ \begin{array}{llll} (y-{{ k}})^2=4{{ p}}(x-{{ h}}) \\\\ \boxed{4{{ p}}(y-{{ k}})=(x-{{ h}})^2 }\\ \end{array} \qquad \begin{array}{llll} vertex\ ({{ h}},{{ k}})\\\\ {{ p}}=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix} \end{array}\\\\ -------------------------------\\\\ 4{{ p}}(y-{{ k}})=(x-{{ h}})^2\quad \begin{cases} h=0\\ k=0\\ p=-7 \end{cases}\implies 4(-7)(y-0)=(x-0)^2 \\\\\\ -28y=x^2\implies y=-\cfrac{1}{28}x^2$