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The focus of a parabola is [tex](0, -2)[/tex]. The directrix is the line [tex]y = 0[/tex]. What is the equation of the parabola in vertex form?

In the equation [tex]y = \frac{1}{4p}(x - k)^2 + h[/tex], the value of [tex]p[/tex] is [tex]\square[/tex].

The vertex of the parabola is the point [tex](\square, -1)[/tex].

The equation of this parabola in vertex form is [tex]y = \square x^2 - 1[/tex].


Sagot :

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].