Question

determine the standard form for the equation of the parabola with vertex (0,0) and focus (0, - 3). include a multiplication sign between symbols. for example, a × π.

Explanation:

Step1: Identify the orientation

Since the vertex is $(0,0)$ and the focus is $(0, - 3)$ (vertex and focus have the same x - coordinate), the parabola opens downwards. The standard - form equation for a parabola opening downwards with vertex $(h,k)$ is $(x - h)^2=4p(y - k)$.

Step2: Determine the value of $p$

The distance between the vertex $(0,0)$ and the focus $(0,-3)$ is given by $|p|$. The formula for the distance between the vertex $(h,k)$ and the focus $(h,k + p)$ is $|p|$. Here, $h = 0,k = 0,k + p=-3$, so $p=-3$.

Step3: Substitute values into the formula

Substitute $h = 0,k = 0,p=-3$ into the equation $(x - h)^2=4p(y - k)$. We get $(x - 0)^2=4\times(-3)(y - 0)$.

Step4: Simplify the equation

$x^{2}=-12y$.

Answer:

$x^{2}=-12y$