A Parabola Has A Focus Located At $\left(5, -\frac{2}{8}\right$\] And A Directrix Of $y = -\frac{7}{8}$. What Are The Coordinates For The Vertex? (____, ____)The Equation For This Parabola In Vertex Form Is As Follows: $y =

Introduction

In the realm of mathematics, particularly in the study of conic sections, parabolas play a vital role. A parabola is a curve that results from the intersection of a cone and a plane. It has several key components, including the focus, directrix, and vertex. In this article, we will delve into the world of parabolas and explore how to find the coordinates of the vertex given the focus and directrix.

Understanding the Focus and Directrix

The focus of a parabola is a fixed point that lies on the axis of symmetry. It is the point from which the parabola is reflected to create its shape. On the other hand, the directrix is a line that is perpendicular to the axis of symmetry and is located at a fixed distance from the focus. The directrix serves as a reference line for the parabola, and the distance between the focus and the directrix is known as the focal length.

The Equation of a Parabola in Vertex Form

The equation of a parabola in vertex form is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the shape of the parabola. The vertex form of a parabola is useful for finding the coordinates of the vertex, as well as for graphing the parabola.

Finding the Vertex of a Parabola

To find the vertex of a parabola, we need to know the coordinates of the focus and the equation of the directrix. The focus is given as $\left(5, -\frac{2}{8}\right)$, and the directrix is given as $y = -\frac{7}{8}$. We can use this information to find the coordinates of the vertex.

Step 1: Determine the Distance Between the Focus and the Directrix

The distance between the focus and the directrix is equal to the focal length. We can find the focal length by subtracting the y-coordinate of the focus from the y-coordinate of the directrix.

$$\text{Focal length} = \left|-\frac{7}{8} - \left(-\frac{2}{8}\right)\right| = \left|-\frac{7}{8} + \frac{2}{8}\right| = \left|-\frac{5}{8}\right| = \frac{5}{8}$$

Step 2: Determine the Coordinates of the Vertex

Since the focus is located at $\left(5, -\frac{2}{8}\right)$, and the directrix is given as $y = -\frac{7}{8}$, we can conclude that the vertex is located at the midpoint between the focus and the directrix. The x-coordinate of the vertex is the same as the x-coordinate of the focus, which is 5. The y-coordinate of the vertex is the average of the y-coordinates of the focus and the directrix.

$$\text{y-coordinate of vertex} = \frac{-\frac{2}{8} + \left(-\frac{7}{8}\right)}{2} = \frac{-\frac{9}{8}}{2} = -\frac{9}{16}$$

Step 3: Write the Equation of the Parabola in Vertex Form

Now that we have the coordinates of the vertex, we can write the equation of the parabola in vertex form.

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the shape of the parabola. Since the vertex is located at $\left(5, -\frac{9}{16}\right)$, we can substitute these values into the equation.

$$y = a(x - 5)^2 - \frac{9}{16}$$

Conclusion

In this article, we explored how to find the coordinates of the vertex of a parabola given the focus and directrix. We used the equation of a parabola in vertex form and the concept of the focal length to determine the coordinates of the vertex. The vertex of a parabola is a critical point that lies on the axis of symmetry, and it plays a vital role in the study of conic sections.

Final Answer

The coordinates of the vertex are $\boxed{\left(5, -\frac{9}{16}\right)}$.

Additional Resources

For more information on parabolas and conic sections, please refer to the following resources:

• Wikipedia: Parabola
• Math Open Reference: Parabola
• Khan Academy: Conic Sections

Related Topics

• Equation of a Parabola
• Focus and Directrix of a Parabola
• Vertex Form of a Parabola

Equation of a Parabola

The equation of a parabola can be written in several forms, including the standard form, the vertex form, and the general form.

Standard Form

The standard form of a parabola is given by:

$$y = ax^2 + bx + c$$

where $a$, $b$, and $c$ are constants that determine the shape and position of the parabola.

Vertex Form

The vertex form of a parabola is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the shape of the parabola.

General Form

The general form of a parabola is given by:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

where $A$, $B$, $C$, $D$, $E$, and $F$ are constants that determine the shape and position of the parabola.

Focus and Directrix of a Parabola

The focus of a parabola is a fixed point that lies on the axis of symmetry. It is the point from which the parabola is reflected to create its shape. The directrix is a line that is perpendicular to the axis of symmetry and is located at a fixed distance from the focus.

Vertex Form of a Parabola

The vertex form of a parabola is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the shape of the parabola.

Conclusion

In this article, we explored the concept of a parabola and its various forms, including the standard form, the vertex form, and the general form. We also discussed the focus and directrix of a parabola and how to find the coordinates of the vertex given the focus and directrix.

Introduction

In our previous article, we explored the concept of a parabola and its various forms, including the standard form, the vertex form, and the general form. We also discussed the focus and directrix of a parabola and how to find the coordinates of the vertex given the focus and directrix. In this article, we will answer some of the most frequently asked questions about parabolas and their vertices.

Q&A

Q: What is the difference between the focus and the vertex of a parabola?

A: The focus of a parabola is a fixed point that lies on the axis of symmetry. It is the point from which the parabola is reflected to create its shape. The vertex, on the other hand, is the point on the parabola that is closest to the focus.

Q: How do I find the coordinates of the vertex of a parabola given the focus and directrix?

A: To find the coordinates of the vertex of a parabola given the focus and directrix, you can use the following steps:

1. Determine the distance between the focus and the directrix.
2. Determine the coordinates of the vertex by finding the midpoint between the focus and the directrix.

Q: What is the equation of a parabola in vertex form?

A: The equation of a parabola in vertex form is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the shape of the parabola.

Q: How do I determine the value of $a$ in the equation of a parabola in vertex form?

A: To determine the value of $a$ in the equation of a parabola in vertex form, you can use the following steps:

1. Find the coordinates of the vertex of the parabola.
2. Substitute the coordinates of the vertex into the equation of the parabola in vertex form.
3. Solve for $a$.

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

A: The vertex of a parabola is a critical point that lies on the axis of symmetry. It plays a vital role in the study of conic sections and is used to determine the shape and position of the parabola.

Q: How do I graph a parabola given its equation in vertex form?

A: To graph a parabola given its equation in vertex form, you can use the following steps:

1. Determine the coordinates of the vertex of the parabola.
2. Substitute the coordinates of the vertex into the equation of the parabola in vertex form.
3. Graph the parabola using the equation.

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

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

• Designing satellite dishes and antennas
• Creating optical systems, such as telescopes and microscopes
• Modeling population growth and decay
• Determining the trajectory of projectiles

Conclusion

In this article, we answered some of the most frequently asked questions about parabolas and their vertices. We discussed the difference between the focus and the vertex of a parabola, how to find the coordinates of the vertex given the focus and directrix, and the significance of the vertex of a parabola. We also provided some real-world applications of parabolas and discussed how to graph a parabola given its equation in vertex form.

Final Answer

The coordinates of the vertex of a parabola are determined by the focus and directrix of the parabola. The equation of a parabola in vertex form is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the shape of the parabola.

Additional Resources

For more information on parabolas and conic sections, please refer to the following resources:

• Wikipedia: Parabola
• Math Open Reference: Parabola
• Khan Academy: Conic Sections

Related Topics

• Equation of a Parabola
• Focus and Directrix of a Parabola
• Vertex Form of a Parabola

Equation of a Parabola

The equation of a parabola can be written in several forms, including the standard form, the vertex form, and the general form.

Standard Form

The standard form of a parabola is given by:

$$y = ax^2 + bx + c$$

where $a$, $b$, and $c$ are constants that determine the shape and position of the parabola.

Vertex Form

The vertex form of a parabola is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the shape of the parabola.

General Form

The general form of a parabola is given by:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

where $A$, $B$, $C$, $D$, $E$, and $F$ are constants that determine the shape and position of the parabola.

Focus and Directrix of a Parabola

The focus of a parabola is a fixed point that lies on the axis of symmetry. It is the point from which the parabola is reflected to create its shape. The directrix is a line that is perpendicular to the axis of symmetry and is located at a fixed distance from the focus.

Vertex Form of a Parabola

The vertex form of a parabola is given by:

$$y = a(x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola, and $a$ is a constant that determines the shape of the parabola.