The Vertex Of The Parabola Below Is At The Point \[$(2, -5)\$\]. Which Of The Equations Below Could Be The One For This Parabola?A. \[$x = -4(y - 2)^2\$\]B. \[$y = (x + 5)^2 + 2\$\]C. \[$y = (x + 2)^2 - 5\$\]D. \[$y
Introduction
In mathematics, a parabola is a type of quadratic equation that can be represented in various forms. One of the key characteristics of a parabola is its vertex, which is the point where the parabola changes direction. The vertex form of a parabola is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this article, we will explore the concept of the vertex of a parabola and how it relates to the equation of the parabola.
Understanding the Vertex Form
The vertex form of a parabola is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. The vertex form is a convenient way to represent a parabola because it allows us to easily identify the vertex of the parabola. The vertex form can be rewritten as y = a(x - h)^2 + k = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + (ah^2 + k).
The Given Parabola
The given parabola has a vertex at the point (2, -5). This means that the equation of the parabola can be written in the form y = a(x - 2)^2 - 5. We are asked to determine which of the given equations could be the one for this parabola.
Analyzing the Options
Let's analyze each of the given options to determine which one could be the equation for the given parabola.
Option A: x = -4(y - 2)^2
This option is not in the standard form of a parabola, and it does not have a vertex at the point (2, -5). Therefore, this option is not a possible equation for the given parabola.
Option B: y = (x + 5)^2 + 2
This option has a vertex at the point (-5, 2), which is not the same as the given vertex (2, -5). Therefore, this option is not a possible equation for the given parabola.
Option C: y = (x + 2)^2 - 5
This option has a vertex at the point (-2, -5), which is not the same as the given vertex (2, -5). However, if we multiply the entire equation by -1, we get y = -(x + 2)^2 + 5, which has a vertex at the point (-2, 5). If we multiply the entire equation by -1 again, we get y = (x + 2)^2 - 5, which has a vertex at the point (-2, -5). Therefore, this option is not a possible equation for the given parabola.
Option D: y = -(x - 2)^2 - 5
This option has a vertex at the point (2, -5), which is the same as the given vertex. Therefore, this option is a possible equation for the given parabola.
Conclusion
In conclusion, the only option that could be the equation for the given parabola is Option D: y = -(x - 2)^2 - 5. This option has a vertex at the point (2, -5), which is the same as the given vertex.
The Final Answer
Q&A: The Vertex of a Parabola
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point where the parabola changes direction. It is the highest or lowest point on the parabola, depending on whether the parabola opens upward or downward.
Q: How do I find the vertex of a parabola?
A: To find the vertex of a parabola, you can use the vertex form of the equation, which is y = a(x - h)^2 + k. The vertex is given by the point (h, k).
Q: What is the vertex form of a parabola?
A: The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Q: How do I determine the equation of a parabola given its vertex?
A: To determine the equation of a parabola given its vertex, you can use the vertex form of the equation. Simply plug in the values of h and k into the equation y = a(x - h)^2 + k.
Q: What is the significance of the vertex of a parabola?
A: The vertex of a parabola is significant because it represents the maximum or minimum value of the function. It is also the point where the parabola changes direction.
Q: Can a parabola have more than one vertex?
A: No, a parabola can only have one vertex. The vertex is a unique point on the parabola.
Q: How do I graph a parabola given its vertex?
A: To graph a parabola given its vertex, you can use the vertex form of the equation. Simply plot the vertex and then use the equation to find other points on the parabola.
Q: What is the relationship between the vertex and the axis of symmetry of a parabola?
A: The vertex of a parabola is the point on the axis of symmetry that is equidistant from the two endpoints of the parabola.
Q: Can a parabola have a vertex at the origin?
A: Yes, a parabola can have a vertex at the origin. This occurs when the equation of the parabola is y = ax^2, where a is a constant.
Q: How do I find the equation of a parabola given its vertex and a point on the parabola?
A: To find the equation of a parabola given its vertex and a point on the parabola, you can use the vertex form of the equation and the point-slope form of a line. Simply plug in the values of the vertex and the point into the equation y = a(x - h)^2 + k.
Conclusion
In conclusion, the vertex of a parabola is a significant point that represents the maximum or minimum value of the function. It is also the point where the parabola changes direction. By understanding the vertex form of a parabola and how to find the equation of a parabola given its vertex, you can graph and analyze parabolas with ease.
The Final Answer
The final answer is: