Find The Vertex Of The Parabola.f(x) = X² - 20x+102OA. (0,10)O B. (2,0)OC. (2,10)OD. (10,2)
Understanding the Problem
To find the vertex of a parabola, we need to understand the properties of a quadratic function. A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants.
In this problem, we are given the quadratic function f(x) = x² - 20x + 102. Our goal is to find the vertex of this parabola, which is the point where the parabola changes direction.
The Vertex Formula
The vertex of a parabola can be found using the formula:
x = -b / 2a
This formula gives us the x-coordinate of the vertex. To find the y-coordinate, we need to plug the x-coordinate back into the original function.
Applying the Formula
Let's apply the formula to the given function f(x) = x² - 20x + 102.
a = 1 (coefficient of x²) b = -20 (coefficient of x)
Now, let's plug these values into the formula:
x = -(-20) / (2 * 1) x = 20 / 2 x = 10
Finding the Y-Coordinate
Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging x = 10 back into the original function:
f(10) = (10)² - 20(10) + 102 f(10) = 100 - 200 + 102 f(10) = 2
Conclusion
Therefore, the vertex of the parabola f(x) = x² - 20x + 102 is (10, 2).
Checking the Options
Let's check the options to see which one matches our answer:
A. (0, 10) - This is not the correct answer. B. (2, 0) - This is not the correct answer. C. (2, 10) - This is not the correct answer. D. (10, 2) - This is the correct answer.
Graphing the Parabola
To visualize the parabola, we can graph it using a graphing tool or a calculator. The graph will show us the shape of the parabola and the location of the vertex.
Real-World Applications
The concept of finding the vertex of a parabola has many real-world applications, such as:
- Optimization problems: Finding the maximum or minimum value of a function is a common problem in optimization.
- Physics and engineering: The vertex of a parabola can represent the maximum height or minimum distance of an object in motion.
- Economics: The vertex of a parabola can represent the maximum profit or minimum cost of a business.
Summary
In this article, we learned how to find the vertex of a parabola using the formula x = -b / 2a. We applied this formula to the given function f(x) = x² - 20x + 102 and found the vertex to be (10, 2). We also checked the options and found that option D is the correct answer. Finally, we discussed the real-world applications of finding the vertex of a parabola.
Frequently Asked Questions
- What is the vertex of a parabola? The vertex of a parabola is the point where the parabola changes direction.
- How do I find the vertex of a parabola? You can use the formula x = -b / 2a to find the x-coordinate of the vertex, and then plug the x-coordinate back into the original function to find the y-coordinate.
- What are the real-world applications of finding the vertex of a parabola?
The concept of finding the vertex of a parabola has many real-world applications, such as optimization problems, physics and engineering, and economics.
Understanding the Problem
To find the vertex of a parabola, we need to understand the properties of a quadratic function. A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants.
Q&A
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point where the parabola changes direction.
Q: How do I find the vertex of a parabola?
A: You can use the formula x = -b / 2a to find the x-coordinate of the vertex, and then plug the x-coordinate back into the original function to find the y-coordinate.
Q: What are the real-world applications of finding the vertex of a parabola?
A: The concept of finding the vertex of a parabola has many real-world applications, such as optimization problems, physics and engineering, and economics.
Q: Can I use the vertex formula to find the vertex of any quadratic function?
A: Yes, the vertex formula can be used to find the vertex of any quadratic function in the form f(x) = ax² + bx + c.
Q: What if the quadratic function is not in the form f(x) = ax² + bx + c?
A: If the quadratic function is not in the form f(x) = ax² + bx + c, you can still use the vertex formula to find the vertex. However, you may need to rewrite the function in the standard form first.
Q: Can I use the vertex formula to find the vertex of a quadratic function with a negative leading coefficient?
A: Yes, the vertex formula can be used to find the vertex of a quadratic function with a negative leading coefficient. However, the vertex will be a maximum point instead of a minimum point.
Q: How do I know if the vertex is a maximum or minimum point?
A: If the leading coefficient (a) is positive, the vertex is a minimum point. If the leading coefficient (a) is negative, the vertex is a maximum point.
Q: Can I use the vertex formula to find the vertex of a quadratic function with a complex coefficient?
A: Yes, the vertex formula can be used to find the vertex of a quadratic function with a complex coefficient. However, the vertex will be a complex number.
Q: What if the quadratic function has a repeated root?
A: If the quadratic function has a repeated root, the vertex will be the same as the root.
Q: Can I use the vertex formula to find the vertex of a quadratic function with a variable coefficient?
A: Yes, the vertex formula can be used to find the vertex of a quadratic function with a variable coefficient. However, the vertex will be a function of the variable.
Conclusion
In this article, we answered some common questions about finding the vertex of a parabola. We discussed the vertex formula, real-world applications, and some special cases. We also provided some examples to illustrate the concepts.
Frequently Asked Questions
- What is the vertex of a parabola? The vertex of a parabola is the point where the parabola changes direction.
- How do I find the vertex of a parabola? You can use the formula x = -b / 2a to find the x-coordinate of the vertex, and then plug the x-coordinate back into the original function to find the y-coordinate.
- What are the real-world applications of finding the vertex of a parabola? The concept of finding the vertex of a parabola has many real-world applications, such as optimization problems, physics and engineering, and economics.
Additional Resources
- Vertex Formula: x = -b / 2a
- Quadratic Function: f(x) = ax² + bx + c
- Real-World Applications: Optimization problems, physics and engineering, and economics.
Summary
In this article, we answered some common questions about finding the vertex of a parabola. We discussed the vertex formula, real-world applications, and some special cases. We also provided some examples to illustrate the concepts.