What Is The Midpoint Of The { X $}$-intercepts Of { F(x) = (x-2)(x-4) $}$?A. { (-3,0)$}$B. { (-1,0)$}$C. { (1,0)$}$D. { (3,0)$}$
Introduction
In mathematics, the x-intercepts of a quadratic function are the points where the graph of the function crosses the x-axis. These points are also known as the roots or solutions of the equation. In this article, we will explore how to find the midpoint of the x-intercepts of a quadratic function.
Understanding Quadratic Functions
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) = ax^2 + bx + c
where a, b, and c are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve.
Finding the x-Intercepts
To find the x-intercepts of a quadratic function, we need to set the function equal to zero and solve for x. This is because the x-intercepts are the points where the graph of the function crosses the x-axis, and at these points, the y-coordinate is zero.
For the given quadratic function f(x) = (x-2)(x-4), we can set it equal to zero and solve for x:
(x-2)(x-4) = 0
This equation can be solved by setting each factor equal to zero:
x - 2 = 0 or x - 4 = 0
Solving for x, we get:
x = 2 or x = 4
Therefore, the x-intercepts of the quadratic function f(x) = (x-2)(x-4) are (2, 0) and (4, 0).
Finding the Midpoint
The midpoint of two points (x1, y1) and (x2, y2) is given by the formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
In this case, we have two points (2, 0) and (4, 0), and we want to find the midpoint of these two points.
Using the formula, we get:
Midpoint = ((2 + 4)/2, (0 + 0)/2) = (6/2, 0) = (3, 0)
Therefore, the midpoint of the x-intercepts of the quadratic function f(x) = (x-2)(x-4) is (3, 0).
Conclusion
In this article, we explored how to find the midpoint of the x-intercepts of a quadratic function. We started by understanding the concept of quadratic functions and how to find the x-intercepts. We then used the formula for finding the midpoint of two points to calculate the midpoint of the x-intercepts of the given quadratic function. The final answer is (3, 0).
Answer
The correct answer is D. (3, 0).
Additional Examples
To further illustrate the concept of finding the midpoint of the x-intercepts of a quadratic function, let's consider a few more examples.
Example 1
Find the midpoint of the x-intercepts of the quadratic function f(x) = (x-1)(x-3).
To find the x-intercepts, we set the function equal to zero and solve for x:
(x-1)(x-3) = 0
This equation can be solved by setting each factor equal to zero:
x - 1 = 0 or x - 3 = 0
Solving for x, we get:
x = 1 or x = 3
Therefore, the x-intercepts of the quadratic function f(x) = (x-1)(x-3) are (1, 0) and (3, 0).
Using the formula for finding the midpoint, we get:
Midpoint = ((1 + 3)/2, (0 + 0)/2) = (4/2, 0) = (2, 0)
Therefore, the midpoint of the x-intercepts of the quadratic function f(x) = (x-1)(x-3) is (2, 0).
Example 2
Find the midpoint of the x-intercepts of the quadratic function f(x) = (x-2)(x-5).
To find the x-intercepts, we set the function equal to zero and solve for x:
(x-2)(x-5) = 0
This equation can be solved by setting each factor equal to zero:
x - 2 = 0 or x - 5 = 0
Solving for x, we get:
x = 2 or x = 5
Therefore, the x-intercepts of the quadratic function f(x) = (x-2)(x-5) are (2, 0) and (5, 0).
Using the formula for finding the midpoint, we get:
Midpoint = ((2 + 5)/2, (0 + 0)/2) = (7/2, 0) = (3.5, 0)
Therefore, the midpoint of the x-intercepts of the quadratic function f(x) = (x-2)(x-5) is (3.5, 0).
Example 3
Find the midpoint of the x-intercepts of the quadratic function f(x) = (x-1)(x-6).
To find the x-intercepts, we set the function equal to zero and solve for x:
(x-1)(x-6) = 0
This equation can be solved by setting each factor equal to zero:
x - 1 = 0 or x - 6 = 0
Solving for x, we get:
x = 1 or x = 6
Therefore, the x-intercepts of the quadratic function f(x) = (x-1)(x-6) are (1, 0) and (6, 0).
Using the formula for finding the midpoint, we get:
Midpoint = ((1 + 6)/2, (0 + 0)/2) = (7/2, 0) = (3.5, 0)
Therefore, the midpoint of the x-intercepts of the quadratic function f(x) = (x-1)(x-6) is (3.5, 0).
Conclusion
Q: What is the midpoint of the x-intercepts of a quadratic function?
A: The midpoint of the x-intercepts of a quadratic function is the point that is equidistant from the two x-intercepts.
Q: How do I find the midpoint of the x-intercepts of a quadratic function?
A: To find the midpoint of the x-intercepts of a quadratic function, you need to follow these steps:
- Find the x-intercepts of the quadratic function by setting the function equal to zero and solving for x.
- Use the formula for finding the midpoint of two points: Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
- Plug in the values of the x-intercepts into the formula and simplify.
Q: What is the formula for finding the midpoint of two points?
A: The formula for finding the midpoint of two points (x1, y1) and (x2, y2) is:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Q: Can I use the midpoint formula to find the midpoint of the x-intercepts of a quadratic function?
A: Yes, you can use the midpoint formula to find the midpoint of the x-intercepts of a quadratic function. However, you need to make sure that the x-intercepts are in the form (x, 0) and not (x, y).
Q: How do I know if the x-intercepts are in the form (x, 0) or (x, y)?
A: To determine if the x-intercepts are in the form (x, 0) or (x, y), you need to look at the y-coordinate of the x-intercepts. If the y-coordinate is 0, then the x-intercepts are in the form (x, 0). If the y-coordinate is not 0, then the x-intercepts are in the form (x, y).
Q: Can I use the midpoint formula to find the midpoint of the x-intercepts of a quadratic function with complex roots?
A: Yes, you can use the midpoint formula to find the midpoint of the x-intercepts of a quadratic function with complex roots. However, you need to make sure that you are using the correct formula for finding the midpoint of complex numbers.
Q: What is the midpoint of the x-intercepts of a quadratic function with complex roots?
A: The midpoint of the x-intercepts of a quadratic function with complex roots is the point that is equidistant from the two complex roots.
Q: How do I find the midpoint of the x-intercepts of a quadratic function with complex roots?
A: To find the midpoint of the x-intercepts of a quadratic function with complex roots, you need to follow these steps:
- Find the complex roots of the quadratic function by using the quadratic formula.
- Use the formula for finding the midpoint of two complex numbers: Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
- Plug in the values of the complex roots into the formula and simplify.
Q: Can I use the midpoint formula to find the midpoint of the x-intercepts of a quadratic function with rational roots?
A: Yes, you can use the midpoint formula to find the midpoint of the x-intercepts of a quadratic function with rational roots. However, you need to make sure that you are using the correct formula for finding the midpoint of rational numbers.
Q: What is the midpoint of the x-intercepts of a quadratic function with rational roots?
A: The midpoint of the x-intercepts of a quadratic function with rational roots is the point that is equidistant from the two rational roots.
Q: How do I find the midpoint of the x-intercepts of a quadratic function with rational roots?
A: To find the midpoint of the x-intercepts of a quadratic function with rational roots, you need to follow these steps:
- Find the rational roots of the quadratic function by using the rational root theorem.
- Use the formula for finding the midpoint of two rational numbers: Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
- Plug in the values of the rational roots into the formula and simplify.
Conclusion
In this article, we have answered some of the most frequently asked questions about finding the midpoint of the x-intercepts of a quadratic function. We have covered topics such as the midpoint formula, complex roots, rational roots, and more. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the concept of finding the midpoint of the x-intercepts of a quadratic function.