Standard Form of a Parabola

Oct 13, 2025 by leolavanderia.com.br 123 views

Introduction

In mathematics, a parabola is a type of quadratic curve that can be represented in various forms, including the standard form, vertex form, and focus-directrix form. The focus-directrix form of a parabola is particularly useful in understanding the properties of the curve, such as the location of the vertex, focus, and directrix. In this article, we will explore how to find the vertex, focus, and directrix of a parabola given in the standard form.

Standard Form of a Parabola

The standard form of a parabola is given by the equation:

y - k = \frac{1}{4p}(x - h)^2

where $(h, k)$ is the vertex of the parabola, and $p$ is the distance between the vertex and the focus. However, the given equation is in the form:

(y-8)^2 = 20(x-9)

Converting to Standard Form

To convert the given equation to the standard form, we need to expand the squared term and rearrange the equation:

(y-8)^2 = 20(x-9)

y^2 - 16y + 64 = 20x - 180

y^2 - 16y - 20x + 64 + 180 = 0

y^2 - 16y - 20x + 244 = 0

Completing the Square

To complete the square, we need to group the $x$ and $y$ terms separately and then add and subtract the square of half the coefficient of the $y$ term:

y^2 - 16y - 20x + 244 = 0

y^2 - 16y + 64 - 64 - 20x + 244 = 0

(y - 8)^2 - 64 - 20x + 244 = 0

(y - 8)^2 - 20x + 180 = 0

Rearranging the Equation

Now, we can rearrange the equation to isolate the squared term:

(y - 8)^2 = 20x - 180

Comparing with Standard Form

Comparing the rearranged equation with the standard form, we can see that:

h = 9

k = 8

p = \frac{1}{20}

Finding the Vertex

The vertex of the parabola is given by the point $(h, k)$ . Therefore, the vertex of the parabola is:

Vertex: $\boxed{(9, 8)}$

Finding the Focus

The focus of the parabola is located at a distance $p$ from the vertex. Since $p = \frac{1}{20}$ , the focus is located at:

Focus: $\boxed{(9 + \frac{1}{20}, 8)}$

Finding the Directrix

The directrix of the parabola is a 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. The equation of the directrix is:

Directrix: $y = 8 - \frac{1}{20}$

Conclusion

In this article, we have explored how to find the vertex, focus, and directrix of a parabola given in the standard form. We have used the standard form of a parabola to convert the given equation to the standard form, completed the square, and then rearranged the equation to isolate the squared term. Finally, we have compared the rearranged equation with the standard form to find the values of $h$ , $k$ , and $p$ . We have then used these values to find the vertex, focus, and directrix of the parabola.

Key Takeaways

The standard form of a parabola is given by the equation $y - k = \frac{1}{4p}(x - h)^2$ .
The vertex of the parabola is given by the point $(h, k)$ .
The focus of the parabola is located at a distance $p$ from the vertex.
The directrix of the parabola is a line that is perpendicular to the axis of symmetry and is located at a distance $p$ from the vertex.

Applications

The focus-directrix form of a parabola has numerous applications in various fields, including physics, engineering, and computer science. For example, the focus-directrix form is used to model the trajectory of projectiles, the motion of objects under the influence of gravity, and the behavior of optical systems.

Future Work

In future work, we can explore other forms of parabolas, such as the vertex form and the parametric form. We can also investigate the properties of parabolas, such as the equation of the tangent line and the equation of the normal line. Additionally, we can apply the focus-directrix form to real-world problems, such as the design of optical systems and the modeling of projectile motion.

References

[1] "Parabolas" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabola.html
[2] "Focus- Directrix Form of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabola2.htm
[3] "Parabolas" by Khan Academy. Retrieved from https://www.khanacademy.org/math/geometry/parabolas/parabolas/v/parabolas

Introduction

In our previous article, we explored the focus-directrix form of a parabola and how to find the vertex, focus, and directrix of a parabola given in the standard form. In this article, we will answer some frequently asked questions about the focus-directrix form of a parabola.

Q&A

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

A: The focus-directrix form of a parabola is a way of representing a parabola in terms of its focus and directrix. It is given by the equation:

y - k = \frac{1}{4p}(x - h)^2

where $(h, k)$ is the vertex of the parabola, and $p$ is the distance between the vertex and the focus.

Q: How do I find the vertex of a parabola in focus-directrix form?

A: To find the vertex of a parabola in focus-directrix form, you need to identify the values of $h$ and $k$ in the equation. The vertex is given by the point $(h, k)$ .

Q: How do I find the focus of a parabola in focus-directrix form?

A: To find the focus of a parabola in focus-directrix form, you need to identify the value of $p$ in the equation. The focus is located at a distance $p$ from the vertex.

Q: How do I find the directrix of a parabola in focus-directrix form?

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

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

A: The focus and directrix of a parabola are related in that they are equidistant from the vertex. The focus is located at a distance $p$ from the vertex, and the directrix is located at a distance $p$ from the vertex.

Q: How do I use the focus-directrix form to model real-world problems?

A: The focus-directrix form can be used to model a variety of real-world problems, such as the motion of projectiles, the behavior of optical systems, and the design of mirrors and lenses.

Q: What are some common applications of the focus-directrix form?

A: The focus-directrix form has numerous applications in various fields, including physics, engineering, and computer science. Some common applications include:

Modeling the trajectory of projectiles
Designing optical systems, such as mirrors and lenses
Modeling the behavior of optical systems, such as telescopes and microscopes
Designing mirrors and lenses for use in optical systems

Q: What are some common mistakes to avoid when working with the focus-directrix form?

A: Some common mistakes to avoid when working with the focus-directrix form include:

Confusing the focus and directrix of a parabola
Failing to identify the values of $h$ , $k$ , and $p$ in the equation
Failing to use the correct formula for finding the vertex, focus, and directrix of a parabola

Conclusion

In this article, we have answered some frequently asked questions about the focus-directrix form of a parabola. We have covered topics such as the definition of the focus-directrix form, how to find the vertex, focus, and directrix of a parabola, and common applications of the focus-directrix form.

Key Takeaways

The focus-directrix form of a parabola is a way of representing a parabola in terms of its focus and directrix.
The vertex of a parabola in focus-directrix form is given by the point $(h, k)$ .
The focus of a parabola in focus-directrix form is located at a distance $p$ from the vertex.
The directrix of a parabola in focus-directrix form is a line that is perpendicular to the axis of symmetry and is located at a distance $p$ from the vertex.
The focus-directrix form has numerous applications in various fields, including physics, engineering, and computer science.

Future Work

References

[1] "Parabolas" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabola.html
[2] "Focus- Directrix Form of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabola2.htm
[3] "Parabolas" by Khan Academy. Retrieved from https://www.khanacademy.org/math/geometry/parabolas/parabolas/v/parabolas