Rewriting the Parabola Equation

Oct 13, 2025 by windailys.com 29 views

Introduction

In mathematics, a parabola is a type of quadratic curve that is defined by a quadratic equation. It is a U-shaped curve that is symmetrical about its axis of symmetry. The focus, directrix, focal diameter, vertex, and axis of symmetry are all important components of a parabola that are used to describe its shape and properties. In this article, we will discuss how to find these components for a given parabola.

The Parabola Equation

The given parabola equation is $30.4y = x^2$ . To find the focus, directrix, focal diameter, vertex, and axis of symmetry, we need to rewrite the equation in the standard form of a parabola, which is $(x - h)^2 = 4p(y - k)$ , where $(h, k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus.

Rewriting the Parabola Equation

To rewrite the equation in the standard form, we need to isolate $y$ on one side of the equation. We can do this by dividing both sides of the equation by $30.4$ :

y = \frac{x^2}{30.4}

Now, we can rewrite the equation as:

\frac{x^2}{30.4} = y

Finding the Vertex

The vertex of a parabola is the point where the parabola changes direction. It is the lowest or highest point on the parabola. To find the vertex, we need to find the values of $h$ and $k$ in the standard form of the parabola.

Since the equation is in the form $y = \frac{x^2}{30.4}$ , we can see that the vertex is at the point $(0, 0)$ .

Finding the Focus

The focus of a parabola is the point that is equidistant from the vertex and the directrix. It is the point where the parabola is tangent to the directrix. To find the focus, we need to find the value of $p$ in the standard form of the parabola.

Since the equation is in the form $y = \frac{x^2}{30.4}$ , we can see that the value of $p$ is $\frac{1}{4 \cdot 30.4} = \frac{1}{121.6}$ .

The focus is at the point $\left(0, \frac{1}{4 \cdot 30.4}\right) = \left(0, \frac{1}{121.6}\right)$ .

Finding the Directrix

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. To find the directrix, we need to find the equation of the line that is perpendicular to the axis of symmetry and is located at a distance $p$ from the vertex.

Since the axis of symmetry is the y-axis, the directrix is a horizontal line that is located at a distance $p$ from the vertex. The equation of the directrix is $y = -\frac{1}{4 \cdot 30.4} = -\frac{1}{121.6}$ .

Finding the Focal Diameter

The focal diameter of a parabola is the distance between the focus and the directrix. To find the focal diameter, we need to find the distance between the focus and the directrix.

The distance between the focus and the directrix is $2p = 2 \cdot \frac{1}{121.6} = \frac{1}{60.8}$ .

Finding the Axis of Symmetry

The axis of symmetry of a parabola is the line that passes through the vertex and is perpendicular to the directrix. To find the axis of symmetry, we need to find the equation of the line that passes through the vertex and is perpendicular to the directrix.

Since the vertex is at the point $(0, 0)$ , the axis of symmetry is the y-axis.

Conclusion

In this article, we discussed how to find the focus, directrix, focal diameter, vertex, and axis of symmetry for a given parabola. We used the standard form of a parabola to find the values of $h$ , $k$ , and $p$ . We also used the values of $h$ , $k$ , and $p$ to find the focus, directrix, focal diameter, vertex, and axis of symmetry.

Key Takeaways

The focus of a parabola is the point that is equidistant from the vertex and the directrix.
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 focal diameter of a parabola is the distance between the focus and the directrix.
The axis of symmetry of a parabola is the line that passes through the vertex and is perpendicular to the directrix.

Final Answer

Introduction

In our previous article, we discussed how to find the focus, directrix, focal diameter, vertex, and axis of symmetry for a given parabola. In this article, we will answer some frequently asked questions about these components of a parabola.

Q: What is the focus of a parabola?

A: The focus of a parabola is the point that is equidistant from the vertex and the directrix. It is the point where the parabola is tangent to the directrix.

Q: How do I find the focus of a parabola?

A: To find the focus of a parabola, you need to find the value of $p$ in the standard form of the parabola. The focus is then located at the point $(0, p)$ .

Q: What is the directrix of a parabola?

A: 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.

Q: How do I find the directrix of a parabola?

A: To find the directrix of a parabola, you need to find the equation of the line that is perpendicular to the axis of symmetry and is located at a distance $p$ from the vertex.

Q: What is the focal diameter of a parabola?

A: The focal diameter of a parabola is the distance between the focus and the directrix.

Q: How do I find the focal diameter of a parabola?

A: To find the focal diameter of a parabola, you need to find the distance between the focus and the directrix. The focal diameter is then equal to $2p$ .

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the lowest or highest point on the parabola.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to find the values of $h$ and $k$ in the standard form of the parabola. The vertex is then located at the point $(h, k)$ .

Q: What is 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.

Q: How do I find the axis of symmetry of a parabola?

A: To find the axis of symmetry of a parabola, you need to find the equation of the line that passes through the vertex and is perpendicular to the directrix.

Q: What is the relationship between the focus, directrix, and focal diameter of a parabola?

A: The focus, directrix, and focal diameter of a parabola are related by the equation $2p = \text{focal diameter}$ , where $p$ is the distance from the vertex to the focus.

Q: How do I use the focus, directrix, and focal diameter to find the equation of a parabola?

A: To use the focus, directrix, and focal diameter to find the equation of a parabola, you need to use the following steps:

Find the focus of the parabola.
Find the directrix of the parabola.
Find the focal diameter of the parabola.
Use the values of the focus, directrix, and focal diameter to find the equation of the parabola.

Conclusion

In this article, we answered some frequently asked questions about the focus, directrix, focal diameter, vertex, and axis of symmetry of a parabola. We hope that this article has been helpful in understanding these components of a parabola.

Key Takeaways

The focus of a parabola is the point that is equidistant from the vertex and the directrix.
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 focal diameter of a parabola is the distance between the focus and the directrix.
The vertex of a parabola is the point where the parabola changes direction.
The axis of symmetry of a parabola is the line that passes through the vertex and is perpendicular to the directrix.

Final Answer

The focus is $\left(0, \frac{1}{121.6}\right)$ . The directrix is $y = -\frac{1}{121.6}$ . The focal diameter is $\frac{1}{60.8}$ . The vertex is $(0, 0)$ . The axis of symmetry is the y-axis.