The Vertex Of This Parabola Is At $(-2, 5$\]. Which Of The Following Could Be Its Equation?A. $y = 3(x-2)^2 - 5$B. $y = 3(x+2)^2 + 5$C. $y = 3(x+2)^2 - 5$
Introduction
In mathematics, a parabola is a fundamental concept that is used to describe the path of an object under the influence of gravity or other forces. The vertex of a parabola is a crucial point that determines its shape and orientation. In this article, we will explore the concept of the vertex of a parabola and how it is used to determine its equation.
What is the Vertex of a Parabola?
The vertex of a parabola is the highest or lowest point on the curve. It is the point where the parabola changes direction, either from opening upwards to downwards or vice versa. The vertex is denoted by the point (h, k), where h is the x-coordinate and k is the y-coordinate.
The Standard Form of a Parabola
The standard 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 'a' determines the direction and width of the parabola.
The Given Vertex
In this problem, the vertex of the parabola is given as (-2, 5). This means that the x-coordinate of the vertex is -2 and the y-coordinate is 5.
Analyzing the Options
Now, let's analyze the given options to determine which one could be the equation of the parabola.
Option A: y = 3(x-2)^2 - 5
In this option, the x-coordinate of the vertex is 2, not -2. Therefore, this option is not correct.
Option B: y = 3(x+2)^2 + 5
In this option, the x-coordinate of the vertex is -2, which matches the given vertex. However, the y-coordinate of the vertex is 5, but the equation has a positive value. This means that the parabola opens upwards, which is not the case.
Option C: y = 3(x+2)^2 - 5
In this option, the x-coordinate of the vertex is -2, which matches the given vertex. The y-coordinate of the vertex is also 5, which matches the given vertex. This means that the parabola opens downwards, which is the case.
Conclusion
Based on the analysis, the correct option is C: y = 3(x+2)^2 - 5. This equation represents a parabola with a vertex at (-2, 5) and opens downwards.
Understanding the Equation
The equation y = 3(x+2)^2 - 5 can be broken down into three parts:
- The vertex form of the parabola is (h, k) = (-2, 5).
- The value of 'a' is 3, which determines the direction and width of the parabola.
- The parabola opens downwards, which means that the value of 'a' is negative.
Graphing the Parabola
To graph the parabola, we can use the equation y = 3(x+2)^2 - 5. We can start by plotting the vertex at (-2, 5). Then, we can use the value of 'a' to determine the direction and width of the parabola.
Real-World Applications
The concept of the vertex of a parabola has many real-world applications. For example, in physics, the vertex of a parabola can be used to describe the path of an object under the influence of gravity. In engineering, the vertex of a parabola can be used to design the shape of a bridge or a building.
Conclusion
Introduction
In our previous article, we explored the concept of the vertex of a parabola and how it is used to determine its equation. In this article, we will answer some frequently asked questions about 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 where the parabola changes direction, either from opening upwards to downwards or vice versa.
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 and k is the y-coordinate.
Q: What is the standard form of a parabola?
A: The standard 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 vertex of the parabola and the value of 'a'. The equation of a parabola can be represented in the standard form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Q: What is the significance of the value of 'a' in the equation of a parabola?
A: The value of 'a' determines the direction and width of the parabola. If 'a' is positive, the parabola opens upwards. If 'a' is negative, the parabola opens downwards.
Q: Can the vertex of a parabola be at any point on the curve?
A: No, the vertex of a parabola cannot be at any point on the curve. The vertex is a specific point that determines the shape and orientation of the parabola.
Q: How do I graph a parabola?
A: To graph a parabola, you need to know the equation of the parabola and the vertex of the parabola. You can start by plotting the vertex and then use the equation to determine the direction and width of the parabola.
Q: What are some real-world applications of the vertex of a parabola?
A: The concept of the vertex of a parabola has many real-world applications. For example, in physics, the vertex of a parabola can be used to describe the path of an object under the influence of gravity. In engineering, the vertex of a parabola can be used to design the shape of a bridge or a building.
Conclusion
In conclusion, the vertex of a parabola is a crucial concept that determines its shape and orientation. By understanding the vertex of a parabola, you can determine the equation of the parabola and graph it. The concept of the vertex of a parabola has many real-world applications and is an important tool in mathematics and science.
Frequently Asked Questions
- What is the vertex of a parabola?
- How is the vertex of a parabola represented?
- What is the standard form of a parabola?
- How do I determine the equation of a parabola?
- What is the significance of the value of 'a' in the equation of a parabola?
- Can the vertex of a parabola be at any point on the curve?
- How do I graph a parabola?
- What are some real-world applications of the vertex of a parabola?
Glossary
- Vertex: The highest or lowest point on the curve of a parabola.
- Standard form: The equation of a parabola in the form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
- Value of 'a': The value that determines the direction and width of the parabola.
- Graphing: The process of plotting the points on a coordinate plane to represent a function or a curve.