Recall the Standard Form of a Parabola

Oct 13, 2025 by miodeya.com 115 views

The vertex form of a parabola is a mathematical representation that provides valuable information about the shape and position of the parabola. In this form, the equation of a parabola is given by:

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

where $(h, k)$ represents the coordinates of the vertex of the parabola, and $a$ is a coefficient that determines the direction and width of the parabola.

Given Information

We are given the vertex $\left(-\frac{1}{2}, 3\right)$ and the directrix $z=-\frac{13}{24}$ . Our goal is to determine the values of $a$ , $h$ , and $k$ in the vertex form of the parabola.

Recall the Standard Form of a Parabola

The standard 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 coefficient that determines the direction and width of the parabola.

Relationship Between the Vertex and Directrix

The vertex and directrix of a parabola are related by the equation:

p = \frac{1}{4a}

where $p$ is the distance between the vertex and the focus, and $a$ is the coefficient of the parabola.

Finding the Value of $a$

We can use the given information to find the value of $a$ . The directrix is given by $z=-\frac{13}{24}$ , which means that the distance between the vertex and the directrix is:

p = \frac{13}{24}

Using the equation $p = \frac{1}{4a}$ , we can solve for $a$ :

\frac{1}{4a} = \frac{13}{24}

a = \frac{24}{4 \cdot 13}

a = \frac{6}{13}

Finding the Values of $h$ and $k$

We are given the vertex $\left(-\frac{1}{2}, 3\right)$ , which means that the values of $h$ and $k$ are:

h = -\frac{1}{2}

k = 3

Conclusion

In conclusion, we have found the values of $a$ , $h$ , and $k$ in the vertex form of the parabola. The value of $a$ is $\frac{6}{13}$ , and the values of $h$ and $k$ are $-\frac{1}{2}$ and $3$ , respectively.

Vertex Form of the Parabola

Using the values of $a$ , $h$ , and $k$ that we found, we can write the vertex form of the parabola as:

y = \frac{6}{13}\left(x + \frac{1}{2}\right)^2 + 3

This equation represents the parabola in vertex form, with the vertex at $\left(-\frac{1}{2}, 3\right)$ and the coefficient $a = \frac{6}{13}$ .

Key Takeaways

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 coefficient that determines the direction and width of the parabola.
The directrix and vertex of a parabola are related by the equation $p = \frac{1}{4a}$ , where $p$ is the distance between the vertex and the focus.
We can use the given information to find the values of $a$ , $h$ , and $k$ in the vertex form of the parabola.
The vertex form of the parabola is given by $y = \frac{6}{13}\left(x + \frac{1}{2}\right)^2 + 3$ , with the vertex at $\left(-\frac{1}{2}, 3\right)$ and the coefficient $a = \frac{6}{13}$ .
Vertex Form of a Parabola: Q&A =====================================

In this article, we will continue to explore the vertex form of a parabola and answer some common questions related to this topic.

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is a mathematical representation that provides valuable information about the shape and position of the parabola. It is given by the equation:

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

where $(h, k)$ represents the coordinates of 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 values of $a$ , $h$ , and $k$ in the vertex form of the parabola?

A: To find the values of $a$ , $h$ , and $k$ in the vertex form of the parabola, you need to have the following information:

The coordinates of the vertex $(h, k)$
The equation of the directrix
The value of $p$ , which is the distance between the vertex and the focus

Using this information, you can use the equation $p = \frac{1}{4a}$ to find the value of $a$ . Then, you can use the coordinates of the vertex to find the values of $h$ and $k$ .

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

A: The vertex and directrix of a parabola are related by the equation:

p = \frac{1}{4a}

where $p$ is the distance between the vertex and the focus, and $a$ is the coefficient of the parabola.

Q: How do I use the vertex form of the parabola to graph the parabola?

A: To graph the parabola using the vertex form, you need to follow these steps:

Identify the values of $a$ , $h$ , and $k$ in the vertex form of the parabola.
Plot the vertex $(h, k)$ on the coordinate plane.
Use the value of $a$ to determine the direction and width of the parabola.
Plot the directrix on the coordinate plane.
Use the equation of the parabola to plot the parabola.

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

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

Not using the correct values of $a$ , $h$ , and $k$ in the vertex form of the parabola.
Not plotting the vertex and directrix correctly on the coordinate plane.
Not using the correct equation of the parabola to plot the parabola.

Q: How do I use the vertex form of the parabola to solve problems involving parabolas?

A: To use the vertex form of the parabola to solve problems involving parabolas, you need to follow these steps:

Identify the values of $a$ , $h$ , and $k$ in the vertex form of the parabola.
Use the equation of the parabola to solve the problem.
Plot the parabola on the coordinate plane to visualize the solution.

Conclusion

In conclusion, the vertex form of a parabola is a powerful tool for working with parabolas. By understanding the vertex form and how to use it to solve problems, you can become proficient in working with parabolas and solving problems involving them.

Key Takeaways

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 coefficient that determines the direction and width of the parabola.
The directrix and vertex of a parabola are related by the equation $p = \frac{1}{4a}$ , where $p$ is the distance between the vertex and the focus.
To find the values of $a$ , $h$ , and $k$ in the vertex form of the parabola, you need to have the coordinates of the vertex, the equation of the directrix, and the value of $p$ .
To graph the parabola using the vertex form, you need to plot the vertex and directrix on the coordinate plane and use the equation of the parabola to plot the parabola.
To solve problems involving parabolas using the vertex form, you need to identify the values of $a$ , $h$ , and $k$ in the vertex form of the parabola and use the equation of the parabola to solve the problem.