To determine the equation of the parabola, we need to understand the relationship between the vertex, focus, and the orientation of the parabola. Here’s a step-by-step explanation:
1. Vertex and Focus of the Parabola:
- The vertex of the parabola is given as [tex]\((0,0)\)[/tex].
- The focus of the parabola is along the negative part of the [tex]\(x\)[/tex]-axis. This means the parabola opens to the left.
2. Standard Form of a Parabola:
- For a parabola that opens to the right or left, the standard form of the equation is [tex]\(y^2 = 4px\)[/tex], where [tex]\(p\)[/tex] is the distance from the vertex to the focus.
- If the parabola opens to the right, [tex]\(p\)[/tex] is positive.
- If the parabola opens to the left, [tex]\(p\)[/tex] is negative.
3. Determining the Value of [tex]\(p\)[/tex]:
- Since the focus is along the negative [tex]\(x\)[/tex]-axis, the parabola opens to the left. Thus, [tex]\(p\)[/tex] must be negative.
4. Equation of the Parabola:
- Given that [tex]\(p\)[/tex] is negative, the general form of the equation becomes [tex]\(y^2 = 4px\)[/tex].
- Since [tex]\(4p\)[/tex] is negative, the equation simplifies to [tex]\(y^2 = -kx\)[/tex] for some positive constant [tex]\(k\)[/tex].
5. Matching with Given Options:
- We need to choose the equation that fits the form [tex]\(y^2 = -kx\)[/tex].
6. Evaluating the Options:
- [tex]\(y^2 = x\)[/tex] — This parabola opens to the right, which doesn't match our condition.
- [tex]\(y^2 = -2x\)[/tex] — This parabola opens to the left, which matches our condition as [tex]\(4p = -2\)[/tex].
- [tex]\(x^2 = 4y\)[/tex] — This parabola opens upward, which doesn't match our condition.
- [tex]\(x^2 = -6y\)[/tex] — This parabola opens downward, which doesn't match our condition.
Given the analysis above, the equation that correctly represents a parabola with vertex at [tex]\((0,0)\)[/tex] and focus along the negative part of the [tex]\(x\)[/tex]-axis is:
[tex]\[ y^2 = -2x \][/tex]
