To determine the equation of a parabola that opens downward with its vertex at the origin and its focus at [tex]\((0, -2)\)[/tex], we need to understand the standard properties of parabolas and use the given information.
1. Standard Form of Parabola:
For a parabola with vertex at the origin [tex]\((0, 0)\)[/tex], the standard form is:
[tex]\[
y = ax^2
\][/tex]
If the parabola opens downward, [tex]\(a\)[/tex] will be negative.
2. Relationship Between Focus and Equation:
The formula for the focus of the parabola [tex]\( y = ax^2 \)[/tex] is given by:
[tex]\[
\left(0, \frac{1}{4a}\right)
\][/tex]
In this problem, the focus is given as [tex]\((0, -2)\)[/tex]. Therefore, we set up the relationship:
[tex]\[
\frac{1}{4a} = -2
\][/tex]
3. Solving for [tex]\(a\)[/tex]:
We need to determine the value of [tex]\(a\)[/tex] that satisfies the above relationship:
[tex]\[
\frac{1}{4a} = -2
\][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[
1 = -8a \quad \text{(by multiplying both sides by 4a)}
\][/tex]
[tex]\[
a = -\frac{1}{8} \quad \text{(by dividing both sides by -8)}
\][/tex]
4. Equation of the Parabola:
With [tex]\(a = -\frac{1}{8}\)[/tex], we substitute back into the standard form:
[tex]\[
y = -\frac{1}{8} x^2
\][/tex]
Therefore, the equation that represents a parabola opening downward with vertex at the origin and focus at [tex]\((0, -2)\)[/tex] is:
[tex]\[
y = -\frac{1}{8} x^2
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{y = -\frac{1}{8} x^2}
\][/tex]
