The Vertex Of This Parabola Is At { (-5,4)$}$. Which Of The Following Could Be Its Equation?A. { Y=-(x-5)^2-4$}$B. { Y=-(x+5)^2+4$}$C. { Y=-(x+5)^2-4$}$D. { Y=-(x-5)^2+4$}$
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 highest or lowest point on the curve. 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 can be used to determine its equation.
Understanding the Vertex
The vertex of a parabola is the point at which the parabola changes direction. It is the highest or lowest point on the curve, depending on the direction of the parabola. 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 value of h represents the x-coordinate of the vertex, and the value of k represents the y-coordinate of the vertex.
Given Information
In this problem, we are given that the vertex of the parabola is at (-5, 4). This means that the x-coordinate of the vertex is -5, and the y-coordinate of the vertex is 4.
Equation of a Parabola
The equation of a parabola can be written in various forms, including the standard form y = ax^2 + bx + c, the vertex form y = a(x - h)^2 + k, and the factored form y = a(x - r)(x - s). In this problem, we are asked to determine which of the given equations could be the equation of the parabola with a vertex at (-5, 4).
Analyzing the Options
Let's analyze each of the given options to determine which one could be the equation of the parabola with a vertex at (-5, 4).
Option A: y = -(x - 5)^2 - 4
This equation has a vertex at (5, -4), which is not the same as the given vertex (-5, 4). Therefore, this option is not correct.
Option B: y = -(x + 5)^2 + 4
This equation has a vertex at (-5, 4), which is the same as the given vertex. Therefore, this option is a possible equation of the parabola.
Option C: y = -(x + 5)^2 - 4
This equation has a vertex at (-5, -4), which is not the same as the given vertex (-5, 4). Therefore, this option is not correct.
Option D: y = -(x - 5)^2 + 4
This equation has a vertex at (5, 4), which is not the same as the given vertex (-5, 4). Therefore, this option is not correct.
Conclusion
In conclusion, the only option that could be the equation of the parabola with a vertex at (-5, 4) is option B: y = -(x + 5)^2 + 4. This equation has a vertex at (-5, 4), which is the same as the given vertex.
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 at which the parabola changes direction. It is the highest or lowest point on the curve, depending on the direction of the parabola.
Q: How is the vertex of a parabola represented in the vertex form of a parabola?
A: The vertex of a parabola is represented in the vertex form of a parabola as (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.
Q: What is the vertex form of a parabola?
A: 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.
Q: How can I determine the equation of a parabola with a given vertex?
A: To determine the equation of a parabola with a given vertex, you can use the vertex form of a parabola. Simply substitute the given vertex into the equation y = a(x - h)^2 + k and solve for a.
Q: What is the significance of the vertex of a parabola?
A: The vertex of a parabola is significant because it represents the highest or lowest point on the curve. It is also used to determine the direction of the parabola.
Q: Can the vertex of a parabola be negative?
A: Yes, the vertex of a parabola can be negative. For example, the vertex of the parabola y = -(x + 5)^2 + 4 is at (-5, 4), which is a negative x-coordinate.
Q: Can the vertex of a parabola be a fraction?
A: Yes, the vertex of a parabola can be a fraction. For example, the vertex of the parabola y = (x - 1/2)^2 + 3 is at (1/2, 3), which is a fraction.
Q: How can I graph a parabola with a given vertex?
A: To graph a parabola with a given vertex, you can use the vertex form of a parabola. Simply substitute the given vertex into the equation y = a(x - h)^2 + k and graph the resulting curve.
Q: What is the relationship between the vertex of a parabola and its axis of symmetry?
A: The vertex of a parabola is the point on the axis of symmetry of the parabola. The axis of symmetry is a vertical line that passes through the vertex of the parabola.
Q: Can the axis of symmetry of a parabola be negative?
A: Yes, the axis of symmetry of a parabola can be negative. For example, the axis of symmetry of the parabola y = -(x + 5)^2 + 4 is the vertical line x = -5, which is a negative x-coordinate.
Q: Can the axis of symmetry of a parabola be a fraction?
A: Yes, the axis of symmetry of a parabola can be a fraction. For example, the axis of symmetry of the parabola y = (x - 1/2)^2 + 3 is the vertical line x = 1/2, which is a fraction.
Conclusion
In conclusion, the vertex of a parabola is an important concept in mathematics that represents the highest or lowest point on the curve. It is used to determine the direction of the parabola and is represented in the vertex form of a parabola as (h, k). 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.