A Parabola With A Vertex At { (0,0)$}$ Has A Directrix That Crosses The Negative Part Of The { Y$}$-axis.Which Could Be The Equation Of The Parabola?A. { X^2 = -4y$}$B. { X^2 = 4y$}$C. { Y^2 = 4x$}$D.

Mar 9, 2025 by ADMIN 201 views

**Understanding Parabolas and Their Equations**

A parabola is a fundamental concept in mathematics, particularly in geometry and algebra. It is a type of curve that is U-shaped and has a vertex, which is the point where the curve changes direction. In this article, we will explore the properties of parabolas and how to determine their equations.

What is a Parabola?

A parabola is a quadratic curve that can be defined as the set of all points that are equidistant to a fixed point called the focus and a fixed line called the directrix. The vertex of a parabola is the point where the curve changes direction, and it is the lowest or highest point on the curve.

Properties of Parabolas

There are several properties of parabolas that are important to understand when working with them. These properties include:

Vertex: The vertex of a parabola is the point where the curve changes direction. It is the lowest or highest point on the curve.
Focus: The focus of a parabola is a fixed point that is used to define the curve. It is the point that is equidistant to all points on the curve.
Directrix: The directrix of a parabola is a fixed line that is used to define the curve. It is the line that is perpendicular to the axis of symmetry of the parabola.
Axis of Symmetry: The axis of symmetry of a parabola is the line that passes through the vertex and is perpendicular to the directrix.

Equations of Parabolas

The equation of a parabola can be written in several forms, including:

Standard Form: The standard form of a parabola is given by the equation $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.
Vertex Form: 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.
Focus-Directrix Form: The focus-directrix form of a parabola is given by the equation $y = \frac{1}{4p}(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus.

Determining the Equation of a Parabola

To determine the equation of a parabola, we need to know the coordinates of the vertex and the distance from the vertex to the focus. We can use this information to write the equation of the parabola in one of the forms mentioned above.

Example: A Parabola with a Vertex at (0,0)

Let's consider a parabola with a vertex at $(0,0)$ . We know that the vertex is the point where the curve changes direction, and it is the lowest or highest point on the curve. We also know that the directrix of the parabola crosses the negative part of the $y$ -axis.

Step 1: Determine the Equation of the Directrix

Since the directrix of the parabola crosses the negative part of the $y$ -axis, we can write the equation of the directrix as $y = -k$ , where $k$ is a positive constant.

Step 2: Determine the Equation of the Parabola

We know that the parabola is symmetric about the $y$ -axis, so the equation of the parabola can be written in the form $x^2 = 4py$ , where $p$ is the distance from the vertex to the focus.

Step 3: Determine the Value of p

We know that the directrix of the parabola crosses the negative part of the $y$ -axis, so the distance from the vertex to the focus is equal to the distance from the vertex to the directrix. We can write this as $p = k$ .

Step 4: Write the Equation of the Parabola

Now that we know the value of $p$ , we can write the equation of the parabola as $x^2 = 4ky$ . Since the vertex is at $(0,0)$ , we can simplify the equation to $x^2 = 4y$ .

Conclusion

In this article, we have explored the properties of parabolas and how to determine their equations. We have seen that a parabola is a quadratic curve that can be defined as the set of all points that are equidistant to a fixed point called the focus and a fixed line called the directrix. We have also seen that the equation of a parabola can be written in several forms, including the standard form, vertex form, and focus-directrix form. Finally, we have seen how to determine the equation of a parabola with a vertex at $(0,0)$ and a directrix that crosses the negative part of the $y$ -axis.

Answer

The equation of the parabola is $x^2 = 4y$ .

References

[1] "Parabolas" by Math Open Reference. Retrieved 2023-02-20.
[2] "Equations of Parabolas" by Purplemath. Retrieved 2023-02-20.
[3] "Focus-Directrix Form of a Parabola" by Wolfram MathWorld. Retrieved 2023-02-20.
Parabolas: A Comprehensive Q&A Guide =============================================

In our previous article, we explored the properties of parabolas and how to determine their equations. In this article, we will answer some of the most frequently asked questions about parabolas.

Q: What is a parabola?

A: A parabola is a quadratic curve that is U-shaped and has a vertex, which is the point where the curve changes direction.

Q: What are the properties of a parabola?

A: The properties of a parabola include:

Vertex: The vertex of a parabola is the point where the curve changes direction. It is the lowest or highest point on the curve.
Focus: The focus of a parabola is a fixed point that is used to define the curve. It is the point that is equidistant to all points on the curve.
Directrix: The directrix of a parabola is a fixed line that is used to define the curve. It is the line that is perpendicular to the axis of symmetry of the parabola.
Axis of Symmetry: The axis of symmetry of a parabola is the line that passes through the vertex and is perpendicular to the directrix.

Q: What are the different forms of a parabola?

A: The different forms of a parabola include:

Standard Form: The standard form of a parabola is given by the equation $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.
Vertex Form: 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.
Focus-Directrix Form: The focus-directrix form of a parabola is given by the equation $y = \frac{1}{4p}(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus.

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 the vertex and the distance from the vertex to the focus. You can use this information to write the equation of the parabola in one of the forms mentioned above.

Q: What is the equation of a parabola with a vertex at (0,0) and a directrix that crosses the negative part of the y-axis?

A: The equation of a parabola with a vertex at (0,0) and a directrix that crosses the negative part of the y-axis is $x^2 = 4y$ .

Q: What is the focus-directrix form of a parabola?

A: The focus-directrix form of a parabola is given by the equation $y = \frac{1}{4p}(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus.

Q: How do I find the distance from the vertex to the focus of a parabola?

A: To find the distance from the vertex to the focus of a parabola, you need to know the equation of the parabola and the coordinates of the vertex. You can use this information to calculate the distance using the formula $p = \frac{1}{4a}$ , where $a$ is the coefficient of the $x^2$ term in the equation of the parabola.

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

A: The axis of symmetry of a parabola is the line that passes through the vertex and is perpendicular to the directrix. It is an important property of a parabola because it helps to determine the equation of the parabola and the location of the focus and directrix.

Q: Can a parabola have a vertex at (0,0) and a directrix that crosses the positive part of the y-axis?

A: No, a parabola cannot have a vertex at (0,0) and a directrix that crosses the positive part of the y-axis. This is because the directrix of a parabola with a vertex at (0,0) must cross the negative part of the y-axis.

Conclusion

In this article, we have answered some of the most frequently asked questions about parabolas. We have seen that a parabola is a quadratic curve that is U-shaped and has a vertex, which is the point where the curve changes direction. We have also seen that the equation of a parabola can be written in several forms, including the standard form, vertex form, and focus-directrix form. Finally, we have seen how to determine the equation of a parabola with a vertex at (0,0) and a directrix that crosses the negative part of the y-axis.

References

[1] "Parabolas" by Math Open Reference. Retrieved 2023-02-20.
[2] "Equations of Parabolas" by Purplemath. Retrieved 2023-02-20.
[3] "Focus-Directrix Form of a Parabola" by Wolfram MathWorld. Retrieved 2023-02-20.