Confusing the vertex and the focus

Oct 13, 2025 by windailys.com 57 views

Which of the Following Equations Represents a Parabola with a Horizontal Axis of Symmetry, Vertex at (-6, 11), and p = 5?

Understanding Parabolas and Their Equations

A parabola is a type of quadratic equation that can be represented in various forms, including standard, vertex, and general forms. The standard form of a parabola with a horizontal axis of symmetry is given by the equation ${(x - h)^2 = 4p(y - k)}$ , where ${(h, k)}$ is the vertex of the parabola and ${p}$ is the distance from the vertex to the focus. In this case, we are given the vertex as ${(-6, 11)}$ and ${p = 5}$ . Our goal is to determine which of the given equations represents a parabola with these characteristics.

Analyzing the Given Options

Let's analyze each of the given options to determine which one represents a parabola with a horizontal axis of symmetry, vertex at ${(-6, 11)}$ , and ${p = 5}$ .

Option A: (x - 11)^2 = 20(y + 6)

This equation is in the standard form of a parabola, but it does not match the given vertex ${(-6, 11)}$ . The vertex of this parabola is ${(11, -6)}$ , which is not the same as the given vertex. Therefore, this option is not correct.

Option B: (x + 11)^2 = -20(y - 6)

This equation is also in the standard form of a parabola, but it has a negative coefficient for the ${y}$ term, which means that the parabola opens downward. However, the given vertex ${(-6, 11)}$ indicates that the parabola opens upward. Therefore, this option is not correct.

Option C: (y - 11)^2 = 20(x + 6)

This equation is in the standard form of a parabola, and it matches the given vertex ${(-6, 11)}$ . The vertex of this parabola is ${(-6, 11)}$ , which is the same as the given vertex. Additionally, the coefficient of the ${x}$ term is positive, which means that the parabola opens to the right. However, we need to check if the value of ${p}$ is correct.

Option D: (y - 11)^2 = -20(x - 6)

This equation is in the standard form of a parabola, and it matches the given vertex ${(-6, 11)}$ . The vertex of this parabola is ${(-6, 11)}$ , which is the same as the given vertex. Additionally, the coefficient of the ${x}$ term is negative, which means that the parabola opens to the left. However, we need to check if the value of ${p}$ is correct.

Option E: (y - 11)^2 = 20(x + 6)

Option F: (y - 11)^2 = -20(x - 6)

This equation is in the standard form of a parabola, and it matches the given vertex ${(-6, 11)}$ . The vertex of this parabola is ${(-6, 11)}$ , which is the same as the given vertex. Additionally, the coefficient of the ${x}$ term is negative, which means that the parabola opens to the left. We need to check if the value of ${p}$ is correct.

Calculating the Value of p

To determine which option is correct, we need to calculate the value of ${p}$ for each option. The value of ${p}$ can be calculated using the formula ${p = \frac{1}{4a}}$ , where ${a}$ is the coefficient of the ${x}$ or ${y}$ term.

For option C, the value of ${a}$ is 20, so the value of ${p}$ is ${p = \frac{1}{4(20)} = \frac{1}{80}}$ . However, this is not equal to the given value of ${p = 5}$ .

For option D, the value of ${a}$ is -20, so the value of ${p}$ is ${p = \frac{1}{4(-20)} = -\frac{1}{80}}$ . However, this is not equal to the given value of ${p = 5}$ .

For option E, the value of ${a}$ is 20, so the value of ${p}$ is ${p = \frac{1}{4(20)} = \frac{1}{80}}$ . However, this is not equal to the given value of ${p = 5}$ .

For option F, the value of ${a}$ is -20, so the value of ${p}$ is ${p = \frac{1}{4(-20)} = -\frac{1}{80}}$ . However, this is not equal to the given value of ${p = 5}$ .

Conclusion

Based on the analysis of the given options, we can conclude that none of the options represent a parabola with a horizontal axis of symmetry, vertex at ${(-6, 11)}$ , and ${p = 5}$ . However, we can see that option C and option E have the correct vertex, but the value of ${p}$ is not correct. We can also see that option D and option F have the correct vertex, but the value of ${p}$ is not correct.

Final Answer

The final answer is not among the given options. However, we can see that the correct equation is ${(y - 11)^2 = 20(x + 6)}$ , but the value of ${p}$ is not correct. The correct value of ${p}$ is ${p = 5}$ , but this is not equal to the value of ${p}$ in the equation. Therefore, the final answer is not among the given options.

Alternative Solution

An alternative solution is to use the vertex form of a parabola, which is given by the equation ${y = a(x - h)^2 + k}$ , where ${(h, k)}$ is the vertex of the parabola. We can substitute the given values of ${h = -6}$ and ${k = 11}$ into this equation to get ${y = a(x + 6)^2 + 11}$ . We can then substitute the given value of ${p = 5}$ into this equation to get ${y = a(x + 6)^2 + 11}$ , where ${a = \frac{1}{4p} = \frac{1}{4(5)} = \frac{1}{20}}$ . Therefore, the correct equation is ${y = \frac{1}{20}(x + 6)^2 + 11}$ .

Final Answer

The final answer is ${y = \frac{1}{20}(x + 6)^2 + 11}$ .
Q&A: Understanding Parabolas and Their Equations

Q: What is a parabola?

A: A parabola is a type of quadratic equation that can be represented in various forms, including standard, vertex, and general forms. It is a U-shaped curve that can open upward or downward.

Q: What is the standard form of a parabola?

A: The standard form of a parabola is given by the equation ${(x - h)^2 = 4p(y - k)}$ , where ${(h, k)}$ is the vertex of the parabola and ${p}$ is the distance from the vertex to the focus.

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is given by the equation ${y = a(x - h)^2 + k}$ , where ${(h, k)}$ is the vertex of the parabola.

Q: How do I determine the value of p in a parabola?

A: The value of ${p}$ can be calculated using the formula ${p = \frac{1}{4a}}$ , where ${a}$ is the coefficient of the ${x}$ or ${y}$ term.

Q: What is the difference between a parabola with a horizontal axis of symmetry and one with a vertical axis of symmetry?

A: A parabola with a horizontal axis of symmetry has its vertex at the point where the axis of symmetry intersects the parabola, and it opens to the left or right. A parabola with a vertical axis of symmetry has its vertex at the point where the axis of symmetry intersects the parabola, and it opens upward or downward.

Q: How do I graph a parabola?

A: To graph a parabola, you can use the following steps:

Determine the vertex of the parabola.
Determine the direction of the parabola (upward or downward).
Plot the vertex on the graph.
Plot two points on either side of the vertex, one above and one below.
Draw a smooth curve through the points to form the parabola.

Q: What are some common applications of parabolas?

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

Physics: Parabolas are used to describe the motion of objects under the influence of gravity.
Engineering: Parabolas are used in the design of mirrors, lenses, and other optical systems.
Computer Science: Parabolas are used in computer graphics to create smooth curves and surfaces.
Mathematics: Parabolas are used to solve quadratic equations and to study the properties of quadratic functions.

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

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

Confusing the vertex and the focus: Make sure to distinguish between the vertex and the focus of the parabola.
Misinterpreting the direction of the parabola: Make sure to determine the direction of the parabola correctly (upward or downward).
Not using the correct formula: Make sure to use the correct formula for the parabola, such as the standard form or the vertex form.
Not checking the units: Make sure to check the units of the variables and the constants in the equation.

Q: How can I practice working with parabolas?

A: You can practice working with parabolas by:

Solving quadratic equations: Practice solving quadratic equations to find the vertex and the focus of the parabola.
Graphing parabolas: Practice graphing parabolas to visualize the shape and the direction of the parabola.
Using online resources: Use online resources, such as graphing calculators or online graphing tools, to practice working with parabolas.
Working with real-world examples: Practice working with real-world examples of parabolas, such as the motion of objects under the influence of gravity or the design of mirrors and lenses.