To find the equation of the parabola with the given focus [tex]\((0, -2)\)[/tex] and directrix [tex]\(y = 0\)[/tex], let's follow these steps:
1. Identify the vertex of the parabola:
- The vertex of a parabola lies exactly midway between the focus and the directrix. Given that the focus is at [tex]\((0, -2)\)[/tex] and the directrix is the line [tex]\(y = 0\)[/tex], the vertex would be:
[tex]\[
\left( 0, \frac{-2 + 0}{2} \right) = (0, -1)
\][/tex]
2. Determine the value of [tex]\(p\)[/tex]:
- The value of [tex]\(p\)[/tex] is the distance from the vertex to the focus (or from the vertex to the directrix, as both must be equal). Here, the distance between [tex]\((0, -1)\)[/tex] and [tex]\((0, -2)\)[/tex] is 1 unit. Since the focus is below the directrix, [tex]\(p\)[/tex] is negative:
[tex]\[
p = -1
\][/tex]
3. Write the equation in vertex form:
- The general form of the equation of a parabola that opens vertically is:
[tex]\[
y = \frac{1}{4p}(x - h)^2 + k
\][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex, so [tex]\(h = 0\)[/tex] and [tex]\(k = -1\)[/tex].
- Substitute [tex]\(p = -1\)[/tex], [tex]\(h = 0\)[/tex], and [tex]\(k = -1\)[/tex]:
[tex]\[
y = \frac{1}{4(-1)}(x - 0)^2 - 1
\][/tex]
Simplify:
[tex]\[
y = -\frac{1}{4}x^2 - 1
\][/tex]
Hence, the detailed solution for the given parabola is as follows:
- The value of [tex]\(p\)[/tex] is [tex]\(\boxed{-1}\)[/tex].
- The vertex of the parabola is the point [tex]\((\boxed{0}, \boxed{-1})\)[/tex].
- The equation of this parabola in vertex form is:
[tex]\[
y = \boxed{-\frac{1}{4}}x^2 - 1
\][/tex]
