Comparison with Given Options

Oct 13, 2025 by recoveryshake.com 120 views

Introduction

Parabolas are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in algebra, geometry, and calculus. In this article, we will delve into the world of parabolas and explore the equation of a parabola opening upward with a focus at ${$ (9, 27)$}$ and a directrix of ${$ y = 11$}$. We will analyze the given options and determine the correct equation.

What is a Parabola?

A parabola is a quadratic curve that can be defined as the set of all points that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). The parabola can open upward or downward, and its shape is determined by the distance between the focus and the directrix.

Equation of a Parabola

The equation of a parabola can be written in the 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. If the parabola opens upward, the value of ${$ a$}$ is positive.

Focus and Directrix

The focus of a parabola is a fixed point that is equidistant from all points on the parabola. The directrix is a fixed line that is perpendicular to the axis of symmetry of the parabola. The distance between the focus and the directrix is called the focal length.

Given Information

We are given that the focus of the parabola is at ${$ (9, 27)$}$ and the directrix is ${$ y = 11$}$. We need to find the equation of the parabola.

Step 1: Find the Vertex

The vertex of the parabola is the midpoint of the line segment joining the focus and the directrix. Since the directrix is ${$ y = 11$}$, the y-coordinate of the vertex is ${$ \frac{27 + 11}{2} = 19$}$. The x-coordinate of the vertex is the same as the x-coordinate of the focus, which is ${ $9\$}$ . Therefore, the vertex of the parabola is at ${$ (9, 19)$}$.

Step 2: Find the Focal Length

The focal length is the distance between the focus and the directrix. Since the directrix is ${$ y = 11$}$ and the focus is at ${$ (9, 27)$}$, the focal length is ${ $27 - 11 = 16\$}$ .

Step 3: Find the Value of a

The value of ${$ a$}$ is related to the focal length by the equation ${$ a = \frac{1}{4f}$}$, where ${$ f$}$ is the focal length. Plugging in the value of the focal length, we get ${$ a = \frac{1}{4(16)} = \frac{1}{64}$}$.

Step 4: Write the Equation of the Parabola

Now that we have the value of ${$ a$}$, we can write the equation of the parabola in the form ${$ y = a(x - h)^2 + k$}$. Plugging in the values of ${$ a$}$, ${$ h$}$, and ${$ k$}$, we get ${$ y = \frac{1}{64}(x - 9)^2 + 19$}$.

Conclusion

In this article, we have analyzed the given options and determined the correct equation of the parabola opening upward with a focus at ${$ (9, 27)$}$ and a directrix of ${$ y = 11$}$. The correct equation is ${$ f(x) = \frac{1}{64}(x - 9)^2 + 19$}$.

Comparison with Given Options

Let's compare the correct equation with the given options:

Option A: ${$ f(x) = \frac{1}{32}(x + 9)^2 - 19$}$
Option B: ${$ f(x) = \frac{1}{32}(x - 9)^2 + 19$}$

The correct equation is ${$ f(x) = \frac{1}{64}(x - 9)^2 + 19$}$, which is different from both options A and B.

Final Answer

$The final answer is ${$f(x) = \frac{1}{64}(x - 9)^2 + 19\$}$.<br/> **Parabola Q&A: Frequently Asked Questions**$

Q: What is a parabola?

A: A parabola is a quadratic curve that can be defined as the set of all points that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).

Q: What is the focus of a parabola?

A: The focus of a parabola is a fixed point that is equidistant from all points on the parabola.

Q: What is the directrix of a parabola?

A: The directrix of a parabola is a fixed line that is perpendicular to the axis of symmetry of the parabola.

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

A: To find the equation of a parabola, you need to know the coordinates of the focus and the directrix. You can then use the formula ${$ y = a(x - h)^2 + k$}$ to write the equation of the parabola.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the midpoint of the line segment joining the focus and the directrix.

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

A: The focal length is the distance between the focus and the directrix. You can find the focal length by subtracting the y-coordinate of the directrix from the y-coordinate of the focus.

Q: What is the value of a in the equation of a parabola?

A: The value of ${$ a$}$ is related to the focal length by the equation ${$ a = \frac{1}{4f}$}$, where ${$ f$}$ is the focal length.

Q: How do I write the equation of a parabola in the form y = ax^2 + bx + c?

A: To write the equation of a parabola in the form ${$ y = ax^2 + bx + c$}$, you need to expand the equation ${$ y = a(x - h)^2 + k$}$ and simplify it.

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

A: The axis of symmetry of a parabola is a line that passes through the vertex of the parabola and is perpendicular to the directrix.

Q: How do I find the x-intercepts of a parabola?

A: To find the x-intercepts of a parabola, you need to set the equation of the parabola equal to zero and solve for x.

Q: How do I find the y-intercept of a parabola?

A: To find the y-intercept of a parabola, you need to set the equation of the parabola equal to zero and solve for y.

Q: What is the standard form of the equation of a parabola?

A: The standard form of the equation of a parabola is ${$ 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 graph a parabola?

A: To graph a parabola, you need to plot the vertex and the x-intercepts, and then draw a smooth curve through the points.

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

A: Parabolas have many real-world applications, including the design of satellite dishes, the shape of a thrown ball, and the trajectory of a projectile.

Q: How do I use a parabola to model a real-world problem?

A: To use a parabola to model a real-world problem, you need to identify the variables and the relationships between them, and then use the equation of the parabola to solve the problem.

Q: What are some common mistakes to avoid when working with parabolas?

A: Some common mistakes to avoid when working with parabolas include:

Not using the correct equation of the parabola
Not identifying the vertex and the x-intercepts correctly
Not using the correct value of ${$ a$}$
Not graphing the parabola correctly

Q: How do I practice working with parabolas?

A: To practice working with parabolas, you can try the following:

Graphing parabolas
Finding the equation of a parabola
Solving problems involving parabolas
Using parabolas to model real-world problems

Q: What are some resources for learning more about parabolas?

A: Some resources for learning more about parabolas include:

Textbooks and online resources
Video tutorials and online courses
Practice problems and worksheets
Real-world applications and examples