Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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]
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]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.