A Parabola, With Its Vertex At The Origin, Has A Directrix At $y = 3$.Which Statements About The Parabola Are True? Select Two Options.A. The Focus Is Located At $(0, -3$\].B. The Parabola Opens To The Left.C. The $p$ Value

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Introduction

A parabola is a fundamental concept in mathematics, particularly in geometry and algebra. It is a U-shaped curve that can be defined in various ways, including as the set of all points that are equidistant to the focus and the directrix. In this article, we will explore a specific type of parabola with its vertex at the origin and a directrix at $y = 3$. We will examine the properties of this parabola and determine which statements about it are true.

Properties of a Parabola

A parabola is defined as the set of all points $(x, y)$ that satisfy the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The vertex of a parabola is the point at which the parabola changes direction, and it is denoted by the coordinates $(h, k)$. The focus of a parabola is a fixed point that is used to define the parabola, and it is located at a distance $p$ from the vertex along the axis of symmetry.

The Directrix of a Parabola

The directrix of a parabola is a line that is perpendicular to the axis of symmetry and is located at a distance $p$ from the vertex. The directrix is used to define the parabola, and it is the set of all points that are equidistant to the focus and the vertex.

The Parabola with Its Vertex at the Origin and a Directrix at $y = 3$

In this case, the parabola has its vertex at the origin $(0, 0)$ and a directrix at $y = 3$. This means that the parabola opens upwards, and the focus is located at a distance $p$ from the vertex along the $y$-axis.

The Focus of the Parabola

The focus of the parabola is located at a distance $p$ from the vertex along the $y$-axis. Since the directrix is at $y = 3$, the focus must be located at a distance $p$ from the vertex along the $y$-axis, which means that it is located at $(0, -3)$.

The $p$ Value

The $p$ value is the distance from the vertex to the focus. In this case, the $p$ value is equal to the distance from the vertex to the directrix, which is $3$.

Conclusion

In conclusion, the parabola with its vertex at the origin and a directrix at $y = 3$ has the following properties:

• The focus is located at $(0, -3)$.
• The $p$ value is equal to $3$.

Therefore, the two true statements about the parabola are:

• A. The focus is located at $(0, -3)$.
• C. The $p$ value is equal to $3$.

The other options are not true, as the parabola opens upwards, not to the left, and the $p$ value is not equal to $-3$.

Discussion

The parabola with its vertex at the origin and a directrix at $y = 3$ is a specific type of parabola that has a number of interesting properties. The focus is located at $(0, -3)$, and the $p$ value is equal to $3$. This parabola is a good example of how the properties of a parabola can be used to determine its equation and graph.

Applications of Parabolas

Parabolas have a number of applications in mathematics and science, including:

• Optics: Parabolas are used to design lenses and mirrors that can focus light.
• Physics: Parabolas are used to describe the motion of objects under the influence of gravity.
• Engineering: Parabolas are used to design curves for bridges and other structures.

Final Thoughts

In conclusion, the parabola with its vertex at the origin and a directrix at $y = 3$ is a specific type of parabola that has a number of interesting properties. The focus is located at $(0, -3)$, and the $p$ value is equal to $3$. This parabola is a good example of how the properties of a parabola can be used to determine its equation and graph.

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Introduction

In our previous article, we explored the properties of a parabola with its vertex at the origin and a directrix at $y = 3$. We determined that the focus is located at $(0, -3)$ and the $p$ value is equal to $3$. In this article, we will answer some frequently asked questions about this parabola.

Q&A

Q: What is the equation of the parabola?

A: The equation of the parabola is $y = \frac{1}{4p}(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is at $(0, 0)$, so the equation of the parabola is $y = \frac{1}{4p}x^2$.

Q: What is the value of $p$?

A: The value of $p$ is equal to the distance from the vertex to the focus. In this case, the focus is located at $(0, -3)$, so the value of $p$ is equal to $3$.

Q: What is the axis of symmetry of the parabola?

A: The axis of symmetry of the parabola is the line that passes through the vertex and is perpendicular to the directrix. In this case, the axis of symmetry is the $y$-axis.

Q: What is the directrix of the parabola?

A: The directrix of the parabola is the line that is perpendicular to the axis of symmetry and is located at a distance $p$ from the vertex. In this case, the directrix is at $y = 3$.

Q: What is the focus of the parabola?

A: The focus of the parabola is the point that is used to define the parabola. In this case, the focus is located at $(0, -3)$.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the point at which the parabola changes direction. In this case, the vertex is at $(0, 0)$.

Q: What is the shape of the parabola?

A: The shape of the parabola is a U-shaped curve that opens upwards.

Q: What are some applications of parabolas?

A: Parabolas have a number of applications in mathematics and science, including optics, physics, and engineering.

Conclusion

In conclusion, the parabola with its vertex at the origin and a directrix at $y = 3$ is a specific type of parabola that has a number of interesting properties. The focus is located at $(0, -3)$, and the $p$ value is equal to $3$. We hope that this Q&A article has been helpful in answering some of the most frequently asked questions about this parabola.

Additional Resources

For more information about parabolas, including their properties and applications, please see the following resources:

• Math Open Reference: A comprehensive online reference for mathematics, including a section on parabolas.
• Wolfram MathWorld: A comprehensive online reference for mathematics, including a section on parabolas.
• Khan Academy: A free online resource for learning mathematics, including a section on parabolas.

Final Thoughts

In conclusion, the parabola with its vertex at the origin and a directrix at $y = 3$ is a specific type of parabola that has a number of interesting properties. We hope that this Q&A article has been helpful in answering some of the most frequently asked questions about this parabola.