Let's determine which equation represents a parabola that opens upward with a vertex at the origin [tex]\((0,0)\)[/tex] and a focus at [tex]\((0,1)\)[/tex].
1. Understanding Parabolas with a Vertex at the Origin:
A parabola with its vertex at the origin [tex]\((0,0)\)[/tex] and opening upwards follows the standard form:
[tex]\[
y = \frac{1}{4p} x^2
\][/tex]
where [tex]\(p\)[/tex] is the distance from the vertex to the focus.
2. Determining [tex]\(p\)[/tex]:
Given the focus [tex]\((0, 1)\)[/tex], the distance [tex]\(p\)[/tex] from the vertex [tex]\((0,0)\)[/tex] to the focus is 1.
3. Substituting [tex]\(p\)[/tex] in the Equation:
Plugging [tex]\(p = 1\)[/tex] into the standard form:
[tex]\[
y = \frac{1}{4 \cdot 1} x^2
\][/tex]
which simplifies to:
[tex]\[
y = \frac{1}{4} x^2
\][/tex]
4. Identifying the Correct Option:
Among the given options:
- [tex]\(y = -\frac{1}{2} x^2\)[/tex]
- [tex]\(y = \frac{1}{2} x^2\)[/tex]
- [tex]\(y = \frac{1}{4} x^2\)[/tex]
- [tex]\(y = \frac{1}{8} x^2\)[/tex]
The equation [tex]\(\boxed{y = \frac{1}{4} x^2}\)[/tex] matches the derived equation.
Thus, the equation that represents a parabola opening upward with a vertex at the origin and a focus at [tex]\((0,1)\)[/tex] is:
[tex]\[
\boxed{y = \frac{1}{4} x^2}
\][/tex]
