Question

write the equation in vertex form for the parabola with vertex (0, -3) and directrix y = -9. simplify any fractions.

Explanation:

Step1: Recall the formula for the vertex form of a parabola

The vertex form of a parabola with vertex \((h,k)\) is \(y = a(x - h)^2 + k\). Here, \(h = 0\) and \(k=-3\), so the equation becomes \(y=a(x - 0)^2-3=a x^{2}-3\).

Step2: Find the value of \(a\)

The distance \(d\) between the vertex \((0,-3)\) and the directrix \(y = - 9\) is \(d=\vert-3-(-9)\vert=\vert-3 + 9\vert = 6\). For a parabola, \(a=\frac{1}{4d}\). Since \(d = 6\), then \(a=\frac{1}{4\times6}=\frac{1}{24}\).

Answer:

\(y=\frac{1}{24}x^{2}-3\)