Finding the Vertex Form of the Equation

Oct 13, 2025 by windailys.com 61 views

Understanding the Basics of a Parabola

A parabola is a type of quadratic equation that can be represented in various forms, including the standard form, vertex form, and focus-directrix form. In this article, we will focus on finding the vertex form of a parabola given a directrix and a focus. The vertex form of a parabola is given by the equation $x = a(y - k)^2 + h$ , where $(h, k)$ is the vertex of the parabola, and $a$ is a coefficient that determines the direction and width of the parabola.

The Focus-Directrix Form of a Parabola

The focus-directrix form of a parabola is given by the equation $x = \frac{1}{4p}(y - k)^2 + h$ , where $p$ is the distance between the focus and the directrix. In this case, we are given the directrix $x = 6$ and the focus $(3, -5)$ . We need to find the value of $p$ and then use it to determine the vertex form of the equation.

Finding the Value of $p$

To find the value of $p$ , we need to calculate the distance between the focus and the directrix. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . In this case, the distance between the focus $(3, -5)$ and the directrix $x = 6$ is given by:

p = \sqrt{(6 - 3)^2 + (0 - (-5))^2} = \sqrt{9 + 25} = \sqrt{34}

Finding the Vertex Form of the Equation

Now that we have the value of $p$ , we can use it to determine the vertex form of the equation. The vertex form of the equation is given by $x = a(y - k)^2 + h$ , where $(h, k)$ is the vertex of the parabola. Since the directrix is given by $x = 6$ , the vertex of the parabola must be at the point $(6, -5)$ . Therefore, we have $h = 6$ and $k = -5$ .

The value of $a$ can be determined using the formula $a = \frac{1}{4p}$ . Substituting the value of $p$ that we found earlier, we get:

a = \frac{1}{4\sqrt{34}}

Simplifying the Vertex Form of the Equation

Now that we have the values of $a$ , $h$ , and $k$ , we can substitute them into the vertex form of the equation to get:

x = \frac{1}{4\sqrt{34}}(y + 5)^2 + 6

This is the vertex form of the equation of the parabola.

Conclusion

In this article, we have shown how to find the vertex form of a parabola given a directrix and a focus. We used the focus-directrix form of a parabola to determine the value of $p$ , which is the distance between the focus and the directrix. We then used this value to determine the vertex form of the equation. The vertex form of the equation is given by $x = a(y - k)^2 + h$ , where $(h, k)$ is the vertex of the parabola, and $a$ is a coefficient that determines the direction and width of the parabola.

Example Problems

Find the vertex form of the equation of a parabola with a directrix $x = 2$ and a focus $(1, -3)$ .
Find the vertex form of the equation of a parabola with a directrix $x = 4$ and a focus $(2, -1)$ .

Solutions to Example Problems

To find the vertex form of the equation of a parabola with a directrix $x = 2$ and a focus $(1, -3)$ , we need to follow the same steps as before. We first find the value of $p$ using the formula $p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . Substituting the values of the directrix and the focus, we get:

p = \sqrt{(2 - 1)^2 + (0 - (-3))^2} = \sqrt{1 + 9} = \sqrt{10}

Next, we use the value of $p$ to determine the vertex form of the equation. The vertex form of the equation is given by $x = a(y - k)^2 + h$ , where $(h, k)$ is the vertex of the parabola. Since the directrix is given by $x = 2$ , the vertex of the parabola must be at the point $(2, -3)$ . Therefore, we have $h = 2$ and $k = -3$ .

The value of $a$ can be determined using the formula $a = \frac{1}{4p}$ . Substituting the value of $p$ that we found earlier, we get:

a = \frac{1}{4\sqrt{10}}

Substituting the values of $a$ , $h$ , and $k$ into the vertex form of the equation, we get:

x = \frac{1}{4\sqrt{10}}(y + 3)^2 + 2

This is the vertex form of the equation of the parabola.

To find the vertex form of the equation of a parabola with a directrix $x = 4$ and a focus $(2, -1)$ , we need to follow the same steps as before. We first find the value of $p$ using the formula $p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . Substituting the values of the directrix and the focus, we get:

p = \sqrt{(4 - 2)^2 + (0 - (-1))^2} = \sqrt{4 + 1} = \sqrt{5}

Next, we use the value of $p$ to determine the vertex form of the equation. The vertex form of the equation is given by $x = a(y - k)^2 + h$ , where $(h, k)$ is the vertex of the parabola. Since the directrix is given by $x = 4$ , the vertex of the parabola must be at the point $(4, -1)$ . Therefore, we have $h = 4$ and $k = -1$ .

The value of $a$ can be determined using the formula $a = \frac{1}{4p}$ . Substituting the value of $p$ that we found earlier, we get:

a = \frac{1}{4\sqrt{5}}

Substituting the values of $a$ , $h$ , and $k$ into the vertex form of the equation, we get:

x = \frac{1}{4\sqrt{5}}(y + 1)^2 + 4

This is the vertex form of the equation of the parabola.

Conclusion

Frequently Asked Questions

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is a way of writing the equation of a parabola in the form $x = a(y - k)^2 + h$ , where $(h, k)$ is the vertex of the parabola, and $a$ is a coefficient that determines the direction and width of the parabola.

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

A: To find the vertex form of a parabola given a directrix and a focus, you need to follow these steps:

Find the value of $p$ using the formula $p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Use the value of $p$ to determine the vertex form of the equation.
Substitute the values of $a$ , $h$ , and $k$ into the vertex form of the equation.

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

A: The vertex form of a parabola is significant because it provides a way of writing the equation of a parabola in a concise and elegant form. It also provides a way of determining the vertex of the parabola, which is an important point in the graph of the parabola.

Q: Can I use the vertex form of a parabola to find the equation of a parabola given its vertex and a point on the parabola?

A: Yes, you can use the vertex form of a parabola to find the equation of a parabola given its vertex and a point on the parabola. To do this, you need to follow these steps:

Substitute the values of the vertex and the point into the vertex form of the equation.
Solve for the value of $a$ .
Substitute the value of $a$ into the vertex form of the equation.

Q: Can I use the vertex form of a parabola to find the equation of a parabola given its focus and a point on the parabola?

A: Yes, you can use the vertex form of a parabola to find the equation of a parabola given its focus and a point on the parabola. To do this, you need to follow these steps:

Find the value of $p$ using the formula $p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Use the value of $p$ to determine the vertex form of the equation.
Substitute the values of $a$ , $h$ , and $k$ into the vertex form of the equation.

Q: Can I use the vertex form of a parabola to find the equation of a parabola given its directrix and a point on the parabola?

A: Yes, you can use the vertex form of a parabola to find the equation of a parabola given its directrix and a point on the parabola. To do this, you need to follow these steps:

Find the value of $p$ using the formula $p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Use the value of $p$ to determine the vertex form of the equation.
Substitute the values of $a$ , $h$ , and $k$ into the vertex form of the equation.

Q: What are some common applications of the vertex form of a parabola?

A: The vertex form of a parabola has many common applications in mathematics and science. Some of these applications include:

Graphing parabolas: The vertex form of a parabola provides a way of graphing parabolas in a concise and elegant form.
Finding the equation of a parabola: The vertex form of a parabola provides a way of finding the equation of a parabola given its vertex, focus, directrix, or a point on the parabola.
Solving problems involving parabolas: The vertex form of a parabola provides a way of solving problems involving parabolas, such as finding the maximum or minimum value of a parabola.

Q: What are some common mistakes to avoid when using the vertex form of a parabola?

A: Some common mistakes to avoid when using the vertex form of a parabola include:

Not following the correct order of operations: When using the vertex form of a parabola, it is essential to follow the correct order of operations to avoid errors.
Not checking the validity of the equation: Before using the vertex form of a parabola, it is essential to check the validity of the equation to avoid errors.
Not using the correct values for the vertex and the point: When using the vertex form of a parabola, it is essential to use the correct values for the vertex and the point to avoid errors.

Q: How can I practice using the vertex form of a parabola?

A: There are many ways to practice using the vertex form of a parabola, including:

Solving problems involving parabolas: Practice solving problems involving parabolas, such as finding the equation of a parabola given its vertex, focus, directrix, or a point on the parabola.
Graphing parabolas: Practice graphing parabolas using the vertex form of a parabola.
Finding the maximum or minimum value of a parabola: Practice finding the maximum or minimum value of a parabola using the vertex form of a parabola.

Q: What are some common resources for learning about the vertex form of a parabola?

A: Some common resources for learning about the vertex form of a parabola include:

Textbooks: Many textbooks cover the vertex form of a parabola in detail.
Online resources: There are many online resources available for learning about the vertex form of a parabola, including video tutorials and practice problems.
Mathematics courses: Many mathematics courses cover the vertex form of a parabola in detail.

Q: Can I use the vertex form of a parabola to find the equation of a parabola given its asymptotes?

A: Yes, you can use the vertex form of a parabola to find the equation of a parabola given its asymptotes. To do this, you need to follow these steps:

Find the value of $p$ using the formula $p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Use the value of $p$ to determine the vertex form of the equation.
Substitute the values of $a$ , $h$ , and $k$ into the vertex form of the equation.

Q: Can I use the vertex form of a parabola to find the equation of a parabola given its axis of symmetry?

A: Yes, you can use the vertex form of a parabola to find the equation of a parabola given its axis of symmetry. To do this, you need to follow these steps:

Find the value of $p$ using the formula $p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Use the value of $p$ to determine the vertex form of the equation.
Substitute the values of $a$ , $h$ , and $k$ into the vertex form of the equation.

Q: Can I use the vertex form of a parabola to find the equation of a parabola given its vertex and a tangent line?

A: Yes, you can use the vertex form of a parabola to find the equation of a parabola given its vertex and a tangent line. To do this, you need to follow these steps:

Find the value of $p$ using the formula $p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Use the value of $p$ to determine the vertex form of the equation.
Substitute the values of $a$ , $h$ , and $k$ into the vertex form of the equation.

Q: Can I use the vertex form of a parabola to find the equation of a parabola given its focus and a tangent line?

A: Yes, you can use the vertex form of a parabola to find the equation of a parabola given its focus and a tangent line. To do this, you need to follow these steps:

Find the value of $p$ using the formula $p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Use the value of $p$ to determine the vertex form of the equation.
Substitute the values of $a$ , $h$ , and $k$ into the vertex form of the equation.