To find the equation of the parabola given the focus and directrix, let's proceed with the following steps:
1. Determine the value of [tex]\( p \)[/tex]:
[tex]\[ p = \frac{\text{distance from the vertex to the focus or directrix}}{2} \][/tex]
Here, the focus is at [tex]\( (0, -2) \)[/tex] and the directrix is the line [tex]\( y = 0 \)[/tex]. The distance between the focus and the directrix is 2 units.
[tex]\[ p = \frac{-2 - 0}{2} = -1 \][/tex]
So, the value of [tex]\( p \)[/tex] is [tex]\(-1\)[/tex].
2. Determine the vertex of the parabola:
The vertex is located midway between the focus and the directrix. Since the focus is at [tex]\( (0, -2) \)[/tex] and the directrix is [tex]\( y = 0 \)[/tex], the vertex is:
[tex]\[ \left( 0, \frac{-2 + 0}{2} \right) = (0, -1) \][/tex]
3. Formulate the equation of the parabola in vertex form [tex]\( y = \frac{1}{4p}(x - h)^2 + k \)[/tex]:
For the given parabola:
- [tex]\( p = -1 \)[/tex]
- Vertex, [tex]\( (h, k) = (0, -1) \)[/tex]
[tex]\[ y = \frac{1}{4(-1)}(x - 0)^2 + (-1) \][/tex]
[tex]\[ y = -\frac{1}{4}x^2 - 1 \][/tex]
Now, fill in the boxes with the obtained values:
In the equation [tex]\( y = \frac{1}{4p}(x - h)^2 + k \)[/tex], 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]
