If The Vertex Of A Parabola Is { (-2, 3)$}$ And One Point On The Parabola Is { (4, 1)$}$, Identify Another Point On The Parabola. Show Work To Support Your Answer.
If the Vertex of a Parabola is Given, How Can We Find Another Point on the Parabola?
Understanding the Problem
In mathematics, a parabola is a type of curve that is defined by a quadratic equation. The vertex of a parabola is the point at which the curve turns, and it is typically represented by the coordinates (h, k). Given the vertex of a parabola and one point on the parabola, we can use this information to find another point on the parabola.
The Vertex Form of a Parabola
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, and a is a constant that determines the direction and width of the parabola.
Given Information
We are given that the vertex of the parabola is (-2, 3) and one point on the parabola is (4, 1). We can use this information to find the value of a in the vertex form of the parabola.
Finding the Value of a
To find the value of a, we can substitute the given point (4, 1) into the vertex form of the parabola:
1 = a(4 - (-2))^2 + 3
Simplifying the equation, we get:
1 = a(6)^2 + 3
1 = 36a + 3
Subtracting 3 from both sides, we get:
-2 = 36a
Dividing both sides by 36, we get:
a = -2/36
a = -1/18
The Equation of the Parabola
Now that we have found the value of a, we can write the equation of the parabola in vertex form:
y = (-1/18)(x + 2)^2 + 3
Finding Another Point on the Parabola
To find another point on the parabola, we can choose a value of x and substitute it into the equation of the parabola. Let's choose x = 0.
y = (-1/18)(0 + 2)^2 + 3
y = (-1/18)(4)^2 + 3
y = (-1/18)(16) + 3
y = -16/18 + 3
y = -8/9 + 3
y = (-8 + 27)/9
y = 19/9
So, another point on the parabola is (0, 19/9).
Conclusion
In this article, we have shown how to find another point on a parabola given the vertex of the parabola and one point on the parabola. We used the vertex form of a parabola and substituted the given point into the equation to find the value of a. We then used the equation of the parabola to find another point on the parabola. This method can be used to find other points on the parabola by choosing different values of x and substituting them into the equation.
Key Takeaways
- The vertex form of a parabola is given by the equation y = a(x - h)^2 + k.
- The vertex of a parabola is the point at which the curve turns, and it is typically represented by the coordinates (h, k).
- Given the vertex of a parabola and one point on the parabola, we can use this information to find another point on the parabola.
- We can find the value of a in the vertex form of the parabola by substituting the given point into the equation.
- We can use the equation of the parabola to find another point on the parabola by choosing a value of x and substituting it into the equation.
Frequently Asked Questions
- 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.
- Q: How can we find another point on a parabola given the vertex of the parabola and one point on the parabola? A: We can use the vertex form of the parabola and substitute the given point into the equation to find the value of a. We can then use the equation of the parabola to find another point on the parabola.
- Q: What is the value of a in the vertex form of the parabola? A: The value of a in the vertex form of the parabola is -1/18.
References
- [1] "Vertex Form of a Parabola" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabolavertexform.html
- [2] "Parabola" by Khan Academy. Retrieved from <https://www.khanacademy.org/math/algebra/x2f4f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f7f
Q&A: Finding Another Point on a Parabola
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, and a is a constant that determines the direction and width of the parabola.
Q: How can we find another point on a parabola given the vertex of the parabola and one point on the parabola?
A: We can use the vertex form of the parabola and substitute the given point into the equation to find the value of a. We can then use the equation of the parabola to find another point on the parabola.
Q: What is the value of a in the vertex form of the parabola?
A: The value of a in the vertex form of the parabola is -1/18.
Q: How do we find the equation of the parabola?
A: To find the equation of the parabola, we can use the vertex form of the parabola and substitute the given point into the equation to find the value of a. We can then use the equation of the parabola to find another point on the parabola.
Q: What is the process of finding another point on a parabola?
A: The process of finding another point on a parabola involves the following steps:
- Find the value of a in the vertex form of the parabola by substituting the given point into the equation.
- Use the equation of the parabola to find another point on the parabola by choosing a value of x and substituting it into the equation.
Q: What are some common mistakes to avoid when finding another point on a parabola?
A: Some common mistakes to avoid when finding another point on a parabola include:
- Not substituting the given point into the equation to find the value of a.
- Not using the equation of the parabola to find another point on the parabola.
- Not choosing a value of x and substituting it into the equation to find another point on the parabola.
Q: How can we check our work when finding another point on a parabola?
A: We can check our work by substituting the new point into the equation of the parabola and verifying that it satisfies the equation.
Q: What are some real-world applications of finding another point on a parabola?
A: Some real-world applications of finding another point on a parabola include:
- Designing parabolic mirrors and lenses.
- Modeling the trajectory of projectiles.
- Analyzing the motion of objects under the influence of gravity.
Q: Can we find another point on a parabola if we only know the vertex of the parabola?
A: Yes, we can find another point on a parabola if we only know the vertex of the parabola. We can use the vertex form of the parabola and substitute the vertex into the equation to find the value of a. We can then use the equation of the parabola to find another point on the parabola.
Q: Can we find another point on a parabola if we only know one point on the parabola?
A: Yes, we can find another point on a parabola if we only know one point on the parabola. We can use the vertex form of the parabola and substitute the given point into the equation to find the value of a. We can then use the equation of the parabola to find another point on the parabola.
Q: How can we find the vertex of a parabola if we only know one point on the parabola?
A: We can find the vertex of a parabola if we only know one point on the parabola by using the equation of the parabola and substituting the given point into the equation to find the value of a. We can then use the equation of the parabola to find the vertex of the parabola.
Q: Can we find the equation of a parabola if we only know the vertex of the parabola?
A: Yes, we can find the equation of a parabola if we only know the vertex of the parabola. We can use the vertex form of the parabola and substitute the vertex into the equation to find the value of a. We can then use the equation of the parabola to find the equation of the parabola.
Q: Can we find the equation of a parabola if we only know one point on the parabola?
A: Yes, we can find the equation of a parabola if we only know one point on the parabola. We can use the vertex form of the parabola and substitute the given point into the equation to find the value of a. We can then use the equation of the parabola to find the equation of the parabola.
Q: How can we find the vertex of a parabola if we only know the equation of the parabola?
A: We can find the vertex of a parabola if we only know the equation of the parabola by using the vertex form of the parabola and substituting the equation into the vertex form to find the value of a. We can then use the equation of the parabola to find the vertex of the parabola.
Q: Can we find the equation of a parabola if we only know the vertex of the parabola and one point on the parabola?
A: Yes, we can find the equation of a parabola if we only know the vertex of the parabola and one point on the parabola. We can use the vertex form of the parabola and substitute the given point into the equation to find the value of a. We can then use the equation of the parabola to find the equation of the parabola.
Q: Can we find the equation of a parabola if we only know one point on the parabola and the vertex of the parabola?
A: Yes, we can find the equation of a parabola if we only know one point on the parabola and the vertex of the parabola. We can use the vertex form of the parabola and substitute the given point into the equation to find the value of a. We can then use the equation of the parabola to find the equation of the parabola.
Q: How can we find the vertex of a parabola if we only know the equation of the parabola and one point on the parabola?
A: We can find the vertex of a parabola if we only know the equation of the parabola and one point on the parabola by using the vertex form of the parabola and substituting the given point into the equation to find the value of a. We can then use the equation of the parabola to find the vertex of the parabola.
Q: Can we find the equation of a parabola if we only know the vertex of the parabola and the equation of the parabola?
A: Yes, we can find the equation of a parabola if we only know the vertex of the parabola and the equation of the parabola. We can use the vertex form of the parabola and substitute the vertex into the equation to find the value of a. We can then use the equation of the parabola to find the equation of the parabola.
Q: Can we find the equation of a parabola if we only know one point on the parabola and the equation of the parabola?
A: Yes, we can find the equation of a parabola if we only know one point on the parabola and the equation of the parabola. We can use the vertex form of the parabola and substitute the given point into the equation to find the value of a. We can then use the equation of the parabola to find the equation of the parabola.
Q: How can we find the vertex of a parabola if we only know the equation of the parabola and the vertex of the parabola?
A: We can find the vertex of a parabola if we only know the equation of the parabola and the vertex of the parabola by using the vertex form of the parabola and substituting the vertex into the equation to find the value of a. We can then use the equation of the parabola to find the vertex of the parabola.
Q: Can we find the equation of a parabola if we only know the vertex of the parabola and one point on the parabola and the equation of the parabola?
A: Yes, we can find the equation of a parabola if we only know the vertex of the parabola and one point on the parabola and the equation of the parabola. We can use the vertex form of the parabola and substitute the given point into the equation to find the value of a. We can then use the equation of the parabola to find the equation of the parabola.
Q: Can we find the equation of a parabola if we only know one point on the parabola and the vertex of the parabola and the equation of the parabola?
A: Yes, we can find the equation of a