Find The Directrix Of The Parabola Given By The Equation $y=\frac{1}{8} X^2$.A. $y=-2$ B. $y=2$ C. $x=-2$

Feb 26, 2025 by ADMIN 110 views

**Finding the Directrix of a Parabola: A Step-by-Step Guide**

Introduction

In mathematics, a parabola is a type of quadratic curve that can be represented by an equation in the form of $y = ax^2 + bx + c$ . The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and does not touch the parabola. In this article, we will focus on finding the directrix of a parabola given by the equation $y = \frac{1}{8} x^2$ .

Understanding the Equation of a Parabola

The equation $y = \frac{1}{8} x^2$ represents a parabola that opens upwards. The coefficient of $x^2$ is $\frac{1}{8}$ , which is a positive value, indicating that the parabola opens upwards. The axis of symmetry of the parabola is the vertical line that passes through the vertex of the parabola.

Finding the Vertex of the Parabola

To find the vertex of the parabola, we need to find the x-coordinate of the vertex. The x-coordinate of the vertex is given by the formula $x = -\frac{b}{2a}$ . In this case, $a = \frac{1}{8}$ and $b = 0$ , so the x-coordinate of the vertex is $x = -\frac{0}{2 \cdot \frac{1}{8}} = 0$ . Therefore, the vertex of the parabola is at the point $(0, 0)$ .

Finding the Directrix of the Parabola

The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and does not touch the parabola. The equation of the directrix is given by the formula $y = k - \frac{1}{4a}$ , where $k$ is the y-coordinate of the vertex. In this case, $k = 0$ and $a = \frac{1}{8}$ , so the equation of the directrix is $y = 0 - \frac{1}{4 \cdot \frac{1}{8}} = 0 - 2 = -2$ .

Conclusion

In conclusion, the directrix of the parabola given by the equation $y = \frac{1}{8} x^2$ is the line $y = -2$ . This line is perpendicular to the axis of symmetry of the parabola and does not touch the parabola.

Step-by-Step Solution

Here is a step-by-step solution to find the directrix of the parabola:

Find the vertex of the parabola: The x-coordinate of the vertex is given by the formula $x = -\frac{b}{2a}$ . In this case, $a = \frac{1}{8}$ and $b = 0$ , so the x-coordinate of the vertex is $x = -\frac{0}{2 \cdot \frac{1}{8}} = 0$ . Therefore, the vertex of the parabola is at the point $(0, 0)$ .
Find the equation of the directrix: The equation of the directrix is given by the formula $y = k - \frac{1}{4a}$ , where $k$ is the y-coordinate of the vertex. In this case, $k = 0$ and $a = \frac{1}{8}$ , so the equation of the directrix is $y = 0 - \frac{1}{4 \cdot \frac{1}{8}} = 0 - 2 = -2$ .

Common Mistakes to Avoid

Here are some common mistakes to avoid when finding the directrix of a parabola:

Mistake 1: Not finding the vertex of the parabola. The vertex of the parabola is necessary to find the equation of the directrix.
Mistake 2: Not using the correct formula for the equation of the directrix. The formula for the equation of the directrix is $y = k - \frac{1}{4a}$ , where $k$ is the y-coordinate of the vertex.
Mistake 3: Not plugging in the correct values for $a$ and $k$ into the formula for the equation of the directrix. Make sure to plug in the correct values for $a$ and $k$ into the formula for the equation of the directrix.

Real-World Applications

The concept of finding the directrix of a parabola has many real-world applications, including:

Optics: The directrix of a parabola is used in optics to design mirrors and lenses.
Engineering: The directrix of a parabola is used in engineering to design curves and surfaces.
Computer Graphics: The directrix of a parabola is used in computer graphics to design curves and surfaces.

Conclusion

In conclusion, finding the directrix of a parabola is an important concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can find the directrix of a parabola given by the equation $y = \frac{1}{8} x^2$ .

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 of the parabola and does not touch the parabola.

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

A: To find the directrix of a parabola, you need to find the vertex of the parabola and then use the formula $y = k - \frac{1}{4a}$ , where $k$ is the y-coordinate of the vertex and $a$ is the coefficient of $x^2$ in the equation of the parabola.

Q: What is the formula for the equation of the directrix?

A: The formula for the equation of the directrix is $y = k - \frac{1}{4a}$ , where $k$ is the y-coordinate of the vertex and $a$ is the coefficient of $x^2$ in the equation of the parabola.

Q: How do I find the y-coordinate of the vertex?

A: The y-coordinate of the vertex is given by the formula $y = c$ , where $c$ is the constant term in the equation of the parabola.

Q: What is the significance of the directrix of a parabola?

A: The directrix of a parabola is significant because it is used to design curves and surfaces in various fields such as optics, engineering, and computer graphics.

Q: Can the directrix of a parabola be a horizontal line?

A: Yes, the directrix of a parabola can be a horizontal line. This occurs when the coefficient of $x^2$ in the equation of the parabola is positive.

Q: Can the directrix of a parabola be a vertical line?

A: No, the directrix of a parabola cannot be a vertical line. This is because the directrix of a parabola is always perpendicular to the axis of symmetry of the parabola, and the axis of symmetry of a parabola is always a vertical line.

Q: How do I determine the orientation of the directrix of a parabola?

A: To determine the orientation of the directrix of a parabola, you need to examine the equation of the parabola and determine whether the coefficient of $x^2$ is positive or negative. If the coefficient of $x^2$ is positive, the directrix of the parabola is a horizontal line. If the coefficient of $x^2$ is negative, the directrix of the parabola is a vertical line.

Q: Can the directrix of a parabola be a line with a slope?

A: No, the directrix of a parabola cannot be a line with a slope. This is because the directrix of a parabola is always a horizontal or vertical line.

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

A: To find the equation of the directrix of a parabola with a given equation, you need to follow these steps:

Find the vertex of the parabola: The x-coordinate of the vertex is given by the formula $x = -\frac{b}{2a}$ . The y-coordinate of the vertex is given by the formula $y = c$ .
Find the equation of the directrix: The equation of the directrix is given by the formula $y = k - \frac{1}{4a}$ , where $k$ is the y-coordinate of the vertex and $a$ is the coefficient of $x^2$ in the equation of the parabola.

Q: What are some common mistakes to avoid when finding the directrix of a parabola?

A: Some common mistakes to avoid when finding the directrix of a parabola include:

Mistake 1: Not finding the vertex of the parabola. The vertex of the parabola is necessary to find the equation of the directrix.
Mistake 2: Not using the correct formula for the equation of the directrix. The formula for the equation of the directrix is $y = k - \frac{1}{4a}$ , where $k$ is the y-coordinate of the vertex and $a$ is the coefficient of $x^2$ in the equation of the parabola.
Mistake 3: Not plugging in the correct values for $a$ and $k$ into the formula for the equation of the directrix. Make sure to plug in the correct values for $a$ and $k$ into the formula for the equation of the directrix.

Q: How do I check my answer for the directrix of a parabola?

A: To check your answer for the directrix of a parabola, you can use the following steps:

Graph the parabola: Graph the parabola using a graphing calculator or a computer program.
Find the directrix: Find the directrix of the parabola using the formula $y = k - \frac{1}{4a}$ , where $k$ is the y-coordinate of the vertex and $a$ is the coefficient of $x^2$ in the equation of the parabola.
Compare the directrix to the graph: Compare the directrix to the graph of the parabola. If the directrix is a horizontal line, it should be a line that is perpendicular to the axis of symmetry of the parabola and does not touch the parabola. If the directrix is a vertical line, it should be a line that is perpendicular to the axis of symmetry of the parabola and does not touch the parabola.

Q: What are some real-world applications of the directrix of a parabola?

A: Some real-world applications of the directrix of a parabola include:

Optics: The directrix of a parabola is used in optics to design mirrors and lenses.
Engineering: The directrix of a parabola is used in engineering to design curves and surfaces.
Computer Graphics: The directrix of a parabola is used in computer graphics to design curves and surfaces.

Q: Can the directrix of a parabola be used to design curves and surfaces in other fields?

A: Yes, the directrix of a parabola can be used to design curves and surfaces in other fields, including architecture, art, and design.