Which Equation Describes A Parabola That Opens Up Or Down And Whose Vertex Is At The Point { (h, V)$}$?A. { Y = A(x - V)^2 + H$}$ B. { X = A(y - V)^2 + H$}$ C. { Y = A(x - H)^2 + V$}$ D. [$x = A(y - H)^2

Introduction

In mathematics, a parabola is a type of quadratic curve that can open up or down. It is a fundamental concept in algebra and geometry, and understanding its properties is crucial for solving various mathematical problems. In this article, we will focus on identifying the equation that describes a parabola that opens up or down and whose vertex is at the point ${(h, v)}$. We will analyze each option and determine which one is the correct equation.

What is a Parabola?

A parabola is a quadratic curve that can be represented by the equation ${y = ax^2 + bx + c}$. It can open up or down, depending on the value of the coefficient ${a}$. If ${a}$ is positive, the parabola opens up, while if ${a}$ is negative, it opens down.

Vertex Form of a 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. This form is useful for identifying the vertex and the direction of the parabola.

Option A: ${y = a(x - v)^2 + h}$

Option A is ${y = a(x - v)^2 + h}$. This equation represents a parabola that opens up or down, depending on the value of ${a}$. However, the vertex of the parabola is at the point ${(v, h)}$, not ${(h, v)}$. Therefore, this option is not correct.

Option B: ${x = a(y - v)^2 + h}$

Option B is ${x = a(y - v)^2 + h}$. This equation represents a parabola that opens left or right, depending on the value of ${a}$. However, the vertex of the parabola is at the point ${(h, v)}$, not ${(h, v)}$. Therefore, this option is not correct.

Option C: ${y = a(x - h)^2 + v}$

Option C is ${y = a(x - h)^2 + v}$. This equation represents a parabola that opens up or down, depending on the value of ${a}$. The vertex of the parabola is at the point ${(h, v)}$, which matches the given condition. Therefore, this option is a strong candidate for the correct equation.

Option D: ${x = a(y - h)^2 + v}$

Option D is ${x = a(y - h)^2 + v}$. This equation represents a parabola that opens left or right, depending on the value of ${a}$. However, the vertex of the parabola is at the point ${(h, v)}$, not ${(h, v)}$. Therefore, this option is not correct.

Conclusion

Based on our analysis, we can conclude that the correct equation that describes a parabola that opens up or down and whose vertex is at the point ${(h, v)}$ is Option C: ${y = a(x - h)^2 + v}$. This equation represents a parabola that opens up or down, depending on the value of ${a}$, and its vertex is at the point ${(h, v)}$, which matches the given condition.

Understanding the Properties of Parabolas

Parabolas are fundamental curves in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we have focused on identifying the equation that describes a parabola that opens up or down and whose vertex is at the point ${(h, v)}$. We have analyzed each option and determined that Option C: ${y = a(x - h)^2 + v}$ is the correct equation.

Key Takeaways

• A parabola is a type of quadratic curve that can open up or down.
• The vertex form of a parabola is given by the equation ${y = a(x - h)^2 + k}$.
• The correct equation that describes a parabola that opens up or down and whose vertex is at the point ${(h, v)}$ is Option C: ${y = a(x - h)^2 + v}$.

Real-World Applications of Parabolas

Parabolas have numerous real-world applications, including:

• Physics: Parabolas are used to describe the trajectory of projectiles, such as thrown balls or launched rockets.
• Engineering: Parabolas are used in the design of bridges, arches, and other structures.
• Computer Science: Parabolas are used in computer graphics to create smooth curves and shapes.

Conclusion

Introduction

In our previous article, we discussed the properties of parabolas and identified the correct equation that describes a parabola that opens up or down and whose vertex is at the point ${(h, v)}$. In this article, we will answer some frequently asked questions about parabolas.

Q: What is a parabola?

A: A parabola is a type of quadratic curve that can open up or down. It is a fundamental concept in algebra and geometry, and understanding its properties is crucial for solving various mathematical problems.

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: What is the equation that describes a parabola that opens up or down and whose vertex is at the point ${(h, v)}$?

A: The correct equation that describes a parabola that opens up or down and whose vertex is at the point ${(h, v)}$ is Option C: ${y = a(x - h)^2 + v}$.

Q: What are some real-world applications of parabolas?

A: Parabolas have numerous real-world applications, including:

• Physics: Parabolas are used to describe the trajectory of projectiles, such as thrown balls or launched rockets.
• Engineering: Parabolas are used in the design of bridges, arches, and other structures.
• Computer Science: Parabolas are used in computer graphics to create smooth curves and shapes.

Q: How do I graph a parabola?

A: To graph a parabola, you can use the following steps:

1. Identify the vertex: The vertex of the parabola is given by the equation ${(h, k)}$.
2. Determine the direction: The parabola opens up or down, depending on the value of ${a}$.
3. Plot the vertex: Plot the vertex on the coordinate plane.
4. Plot the axis of symmetry: The axis of symmetry is a vertical line that passes through the vertex.
5. Plot the parabola: Plot the parabola by using the equation ${y = a(x - h)^2 + k}$.

Q: How do I find the equation of a parabola given its graph?

A: To find the equation of a parabola given its graph, you can use the following steps:

1. Identify the vertex: The vertex of the parabola is the point where the parabola changes direction.
2. Determine the direction: The parabola opens up or down, depending on the value of ${a}$.
3. Find the equation: Use the equation ${y = a(x - h)^2 + k}$ to find the equation of the parabola.

Q: What are some common mistakes to avoid when working with parabolas?

A: Some common mistakes to avoid when working with parabolas include:

• Confusing the vertex and the axis of symmetry: The vertex is the point where the parabola changes direction, while the axis of symmetry is a vertical line that passes through the vertex.
• Using the wrong equation: Make sure to use the correct equation for the parabola, such as ${y = a(x - h)^2 + k}$.
• Not considering the direction: Make sure to consider the direction of the parabola, which is determined by the value of ${a}$.

Conclusion

In conclusion, parabolas are fundamental curves in mathematics, and understanding their properties is crucial for solving various mathematical problems. We have answered some frequently asked questions about parabolas, including their definition, vertex form, and real-world applications. We hope that this article has been helpful in clarifying any doubts you may have had about parabolas.