The Vertex Of This Parabola Is At { (4,-3)$}$. Which Of The Following Could Be Its Equation?A. { X = -3(y-4)^2 - 3$}$B. { X = -3(y+3)^2 + 4$}$C. { X = -3(y+4)^2 - 3$}$
Introduction
In mathematics, a parabola is a type of quadratic equation that can be represented in various forms. One of the most important aspects 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 to determine its equation.
Understanding the Vertex of a Parabola
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.
Determining the Equation of a Parabola
To determine the equation of a parabola, we need to know the coordinates of its vertex. In this case, the vertex of the parabola is given as (4, -3). We can use this information to determine the equation of the parabola.
Option A: x = -3(y-4)^2 - 3
Let's analyze the first option: x = -3(y-4)^2 - 3. In this equation, the vertex form is not explicitly given. However, we can rewrite the equation in vertex form by expanding the squared term:
x = -3(y^2 - 8y + 16) - 3
x = -3y^2 + 24y - 48 - 3
x = -3y^2 + 24y - 51
Comparing this equation with the standard vertex form y = a(x - h)^2 + k, we can see that the equation is not in the correct form. The x-coordinate of the vertex is not 4, and the y-coordinate of the vertex is not -3.
Option B: x = -3(y+3)^2 + 4
Let's analyze the second option: x = -3(y+3)^2 + 4. In this equation, the vertex form is given, and the vertex is (h, k) = (-3, 4). However, the vertex of the parabola is given as (4, -3), which is not the same as the vertex in this equation.
Option C: x = -3(y+4)^2 - 3
Let's analyze the third option: x = -3(y+4)^2 - 3. In this equation, the vertex form is given, and the vertex is (h, k) = (-4, -3). However, the vertex of the parabola is given as (4, -3), which is not the same as the vertex in this equation.
Conclusion
In conclusion, the correct equation of the parabola is not among the options given. The vertex of the parabola is (4, -3), and the equation of the parabola is not in the correct form in any of the options.
The Correct Equation
To determine the correct equation of the parabola, we need to use the vertex form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this case, the vertex is (4, -3), so the equation is:
y = a(x - 4)^2 - 3
To find the value of a, we can use the fact that the parabola passes through the point (0, -3). Substituting this point into the equation, we get:
-3 = a(0 - 4)^2 - 3
-3 = 16a - 3
16a = 0
a = 0
However, this value of a does not give us a parabola. To get a parabola, we need to choose a different value of a. Let's choose a = -1. Then, the equation of the parabola is:
y = -1(x - 4)^2 - 3
y = -(x^2 - 8x + 16) - 3
y = -x^2 + 8x - 19
This is the correct equation of the parabola.
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 highest or lowest point on the curve. It is the point at which the parabola changes direction.
Q: How is the vertex of a parabola represented?
A: The vertex of a parabola is represented by the point (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 do I determine the equation of a parabola?
A: To determine the equation of a parabola, you need to know the coordinates of its vertex. You can use the vertex form of a parabola to write the equation.
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 the point at which the parabola changes direction.
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 - 4)^2 - 3 is (4, -3).
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 - 2/3)^2 - 1 is (2/3, -1).
Q: How do I find the equation of a parabola given its vertex?
A: To find the equation of a parabola given its vertex, you can use the vertex form of a parabola. For example, if the vertex of the parabola is (4, -3), the equation of the parabola is y = a(x - 4)^2 - 3.
Q: What is the value of a in the equation of a parabola?
A: The value of a in the equation of a parabola determines the direction and shape of the parabola. A positive value of a gives a parabola that opens upward, while a negative value of a gives a parabola that opens downward.
Q: Can the value of a be zero?
A: Yes, the value of a can be zero. However, this would give a parabola that is a straight line, not a parabola.
Q: How do I determine the value of a in the equation of a parabola?
A: To determine the value of a in the equation of a parabola, you can use the fact that the parabola passes through a given point. You can substitute the coordinates of the point into the equation and solve for a.
Q: What is the significance of the axis of symmetry of a parabola?
A: The axis of symmetry of a parabola is a line that passes through the vertex of the parabola and is perpendicular to the directrix of the parabola. It is significant because it represents the line of symmetry 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 - 4)^2 - 3 is x = 4.
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 - 2/3)^2 - 1 is x = 2/3.
Q: How do I find the equation of a parabola given its axis of symmetry?
A: To find the equation of a parabola given its axis of symmetry, you can use the fact that the axis of symmetry passes through the vertex of the parabola. You can use the vertex form of a parabola to write the equation.
Q: What is the significance of the directrix of a parabola?
A: The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of 1/(4a) from the vertex of the parabola. It is significant because it represents the line that is perpendicular to the axis of symmetry of the parabola.
Q: Can the directrix of a parabola be negative?
A: Yes, the directrix of a parabola can be negative. For example, the directrix of the parabola y = -(x - 4)^2 - 3 is y = -1/4.
Q: Can the directrix of a parabola be a fraction?
A: Yes, the directrix of a parabola can be a fraction. For example, the directrix of the parabola y = -(x - 2/3)^2 - 1 is y = -1/12.
Q: How do I find the equation of a parabola given its directrix?
A: To find the equation of a parabola given its directrix, you can use the fact that the directrix is perpendicular to the axis of symmetry of the parabola. You can use the vertex form of a parabola to write the equation.
Conclusion
In conclusion, the vertex of a parabola is a significant concept in mathematics that represents the highest or lowest point on the curve. It is represented by the point (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex. The equation of a parabola can be determined using the vertex form of a parabola, and the value of a determines the direction and shape of the parabola. The axis of symmetry and directrix of a parabola are also significant concepts that represent the line of symmetry and the line that is perpendicular to the axis of symmetry, respectively.