Write The Equation In Vertex Form For The Parabola With Vertex \[$(-1, 0)\$\] And Focus \[$(8, 0)\$\]. Simplify Any Fractions.
Introduction
In mathematics, a parabola is a quadratic curve that can be represented in various forms, including the standard form, vertex form, and focus form. The vertex form of a parabola is a powerful tool for analyzing and graphing parabolic curves. In this article, we will explore the vertex form of a parabola, with a focus on writing the equation in vertex form for a parabola with a given vertex and focus.
What is Vertex Form?
The vertex form of a parabola is a mathematical representation of a parabola in the form:
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.
Vertex Form of a Parabola with Vertex (-1, 0) and Focus (8, 0)
To write the equation in vertex form for a parabola with vertex (-1, 0) and focus (8, 0), we need to use the following formula:
p = (focus - vertex) / 2
where p is the distance between the focus and the vertex.
p = (8 - (-1)) / 2 p = 9 / 2 p = 4.5
Now that we have the value of p, we can use it to write the equation in vertex form:
y = a(x - h)^2 + k
Since the vertex is (-1, 0), we can substitute h = -1 and k = 0 into the equation:
y = a(x + 1)^2
To find the value of 'a', we need to use the fact that the focus is (8, 0). We can use the formula:
p = 1 / (4a)
to solve for 'a':
4.5 = 1 / (4a) a = 1 / (4 * 4.5) a = 1 / 18 a = 1/18
Now that we have the value of 'a', we can substitute it into the equation:
y = (1/18)(x + 1)^2
Simplifying the Equation
To simplify the equation, we can multiply both sides by 18 to eliminate the fraction:
18y = (x + 1)^2
Expanding the Equation
To expand the equation, we can use the formula:
(x + 1)^2 = x^2 + 2x + 1
Substituting this into the equation, we get:
18y = x^2 + 2x + 1
Rearranging the Equation
To rearrange the equation, we can subtract 18y from both sides to isolate the term with the variable:
x^2 + 2x + 1 - 18y = 0
Simplifying the Equation
To simplify the equation, we can combine like terms:
x^2 + 2x - 18y + 1 = 0
Conclusion
In this article, we have explored the vertex form of a parabola and written the equation in vertex form for a parabola with vertex (-1, 0) and focus (8, 0). We have used the formula p = (focus - vertex) / 2 to find the value of p, and then used the formula p = 1 / (4a) to solve for 'a'. We have also simplified and expanded the equation to obtain the final form of the equation in vertex form.
Vertex Form of a Parabola: Key Takeaways
- The vertex form of a parabola is a mathematical representation of a parabola in the form y = a(x - h)^2 + k.
- The vertex form of a parabola can be used to analyze and graph parabolic curves.
- The equation in vertex form can be written using the formula p = (focus - vertex) / 2 and p = 1 / (4a).
- The equation in vertex form can be simplified and expanded to obtain the final form of the equation.
References
- [1] "Vertex Form of a Parabola" by Math Open Reference
- [2] "Vertex Form of a Parabola" by Purplemath
- [3] "Vertex Form of a Parabola" by Khan Academy
Vertex Form of a Parabola: Q&A =====================================
Introduction
In our previous article, we explored the vertex form of a parabola and wrote the equation in vertex form for a parabola with vertex (-1, 0) and focus (8, 0). In this article, we will answer some frequently asked questions about the vertex form of a parabola.
Q: What is the vertex form of a parabola?
A: The vertex form of a parabola is a mathematical representation of a parabola in the form 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.
Q: How do I write the equation in vertex form for a parabola with a given vertex and focus?
A: To write the equation in vertex form for a parabola with a given vertex and focus, you need to use the formula p = (focus - vertex) / 2, where p is the distance between the focus and the vertex. Then, you can use the formula p = 1 / (4a) to solve for 'a'.
Q: What is the significance of the vertex form of a parabola?
A: The vertex form of a parabola is a powerful tool for analyzing and graphing parabolic curves. It allows you to easily identify the vertex, focus, and direction of the parabola, making it a useful tool for solving problems in mathematics and science.
Q: Can I use the vertex form of a parabola to solve problems in real-world applications?
A: Yes, the vertex form of a parabola can be used to solve problems in real-world applications, such as physics, engineering, and economics. For example, you can use the vertex form of a parabola to model the trajectory of a projectile, or to analyze the behavior of a system with a parabolic response.
Q: How do I simplify and expand the equation in vertex form?
A: To simplify and expand the equation in vertex form, you can use algebraic manipulations, such as multiplying both sides by a constant, or expanding the squared term using the formula (x + 1)^2 = x^2 + 2x + 1.
Q: Can I use the vertex form of a parabola to find the equation of a parabola with a given vertex and focus?
A: Yes, you can use the vertex form of a parabola to find the equation of a parabola with a given vertex and focus. Simply substitute the values of the vertex and focus into the formula p = (focus - vertex) / 2, and then use the formula p = 1 / (4a) to solve for 'a'.
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:
- Failing to use the correct formula for the vertex form of a parabola.
- Not simplifying and expanding the equation correctly.
- Not using the correct values for the vertex and focus.
- Not checking the solution for errors.
Conclusion
In this article, we have answered some frequently asked questions about the vertex form of a parabola. We have covered topics such as the significance of the vertex form of a parabola, how to write the equation in vertex form, and how to simplify and expand the equation. We hope that this article has been helpful in clarifying any confusion you may have had about the vertex form of a parabola.
Vertex Form of a Parabola: Key Takeaways
- The vertex form of a parabola is a mathematical representation of a parabola in the form y = a(x - h)^2 + k.
- The vertex form of a parabola can be used to analyze and graph parabolic curves.
- The equation in vertex form can be written using the formula p = (focus - vertex) / 2 and p = 1 / (4a).
- The equation in vertex form can be simplified and expanded to obtain the final form of the equation.
- The vertex form of a parabola can be used to solve problems in real-world applications.
References
- [1] "Vertex Form of a Parabola" by Math Open Reference
- [2] "Vertex Form of a Parabola" by Purplemath
- [3] "Vertex Form of a Parabola" by Khan Academy