Given the focus of the parabola [tex]\((0, -2)\)[/tex] and the directrix [tex]\(y = 0\)[/tex]:
1. Finding [tex]\(p\)[/tex]:
- The distance [tex]\(2p\)[/tex] between the focus and the directrix is the distance from [tex]\((0, -2)\)[/tex] to the line [tex]\(y = 0\)[/tex].
- Thus, [tex]\(2p = 2\)[/tex], so [tex]\(p = 1\)[/tex].
2. Finding the vertex:
- The vertex is halfway between the focus and the directrix.
- The midpoint of the y-coordinates [tex]\(-2\)[/tex] and [tex]\(0\)[/tex] is [tex]\(\frac{-2 + 0}{2} = -1\)[/tex].
- Therefore, the vertex is [tex]\((0, -1)\)[/tex].
3. Equation in vertex form:
- The vertex form of the parabola is [tex]\(y = \frac{1}{4p}(x-h)^2 + k\)[/tex].
- Here, [tex]\(p = 1\)[/tex], [tex]\(h = 0\)[/tex], and [tex]\(k = -1\)[/tex].
- Therefore, the equation is [tex]\(y = \frac{1}{4 \cdot 1}(x - 0)^2 - 1 = \frac{1}{4}(x^2) - 1 = 0.25x^2 - 1\)[/tex].
Filling in the blanks:
- The value of [tex]\(p\)[/tex] is [tex]\(1\)[/tex].
- The vertex of the parabola is the point [tex]\((0, -1)\)[/tex].
- The equation of the parabola in vertex form is [tex]\(y = 0.25x^2 - 1\)[/tex].
