Question

write the equation in vertex form for the parabola with vertex (0, -2) and directrix y = 4. simplify any fractions.

Explanation:

Step1: Find the value of \(p\)

The formula for \(p\) is \(p=\frac{y_{directrix}-y_{vertex}}{2}\).
Substitute \(y_{directrix} = 4\) and \(y_{vertex}=-2\) into the formula:
\(p=\frac{4 - (-2)}{2}=\frac{4 + 2}{2}=3\).
Since the directrix \(y = 4\) is above the vertex \((0,-2)\), the parabola opens downwards, so \(p=- 3\).

Step2: Write the vertex form of the parabola equation

The vertex form of a parabola equation is \(y=a(x - h)^{2}+k\), where \((h,k)\) is the vertex and \(a=\frac{1}{4p}\).
Here \(h = 0,k=-2\) and \(p=-3\).
First, find \(a\): \(a=\frac{1}{4\times(-3)}=-\frac{1}{12}\).
Then the equation is \(y=-\frac{1}{12}(x - 0)^{2}-2\).

Answer:

\(y =-\frac{1}{12}x^{2}-2\)