Question

write an equation of the parabola with vertex at $(0,0)$ and directrix $y = 12$. use the variables $x$ and $y$. an equation of the parabola is $\square$.

Explanation:

Step1: Recall the standard form of parabola equation

For a parabola with vertex \((h,k)\) and directrix \(y = k - p\), the standard form is \((x - h)^2=4p(y - k)\). Here, \(h = 0,k = 0\), and the directrix \(y=12\). Since \(y=k - p\), then \(12=0 - p\), so \(p=- 12\).

Step2: Substitute \(h,k,p\) into the standard form

Substitute \(h = 0,k = 0,p=-12\) into \((x - h)^2=4p(y - k)\). We get \(x^{2}=4\times(-12)y\).

Answer:

\(x^{2}=-48y\)