Question

which is the standard form of the equation of a parabola with a focus of (0, -3) and its vertex at the origin?
$y^2 = 12x$
$x^2 = 12y$
$x^2 = -12y$
$y^2 = -12x$

Explanation:

Step1: Determine parabola direction

The vertex is at \((0,0)\) and focus at \((0, -3)\), so it opens downward (since focus is below vertex). For vertical parabolas, standard form is \(x^2 = 4py\), where \(p\) is the distance from vertex to focus.

Step2: Calculate \(p\)

Focus is \((0, -3)\), so \(p = -3\) (negative as it opens down).

Step3: Substitute \(p\) into formula

Substitute \(p = -3\) into \(x^2 = 4py\): \(x^2 = 4(-3)y = -12y\).

Answer:

\(x^2 = -12y\)