Finding The X-Intercept: F(x) = (x-4)(x+2)

by Alex Johnson 43 views

In this article, we will walk through the process of identifying the x-intercept(s) of the quadratic function f(x) = (x-4)(x+2). Understanding how to find x-intercepts is a fundamental skill in algebra and is crucial for analyzing the behavior of quadratic equations and their corresponding parabolas. We will break down the steps involved and explain the underlying concepts to help you grasp this essential mathematical skill. So, let's dive in and explore how to find those crucial points where the parabola intersects the x-axis.

Understanding X-Intercepts

X-intercepts, also known as roots or zeros of a function, are the points where the graph of the function intersects the x-axis. At these points, the value of the function, f(x), is equal to zero. In the context of a quadratic function, the x-intercepts are the solutions to the equation f(x) = 0. Identifying these points is vital as they provide key information about the quadratic function's behavior, such as where the parabola crosses the x-axis and the symmetry of the graph. To find the x-intercepts, we essentially need to solve the quadratic equation. For the given function f(x) = (x-4)(x+2), this involves setting the function equal to zero and solving for x. This will give us the x-coordinates of the points where the parabola intersects the x-axis. Understanding the significance of x-intercepts not only helps in graphing quadratic functions but also in solving real-world problems that can be modeled using quadratic equations. For instance, in physics, the x-intercepts can represent the time at which a projectile hits the ground, or in business, they can indicate break-even points where profit equals zero.

Steps to Find the X-Intercepts

To find the x-intercepts of the quadratic function f(x) = (x-4)(x+2), we need to follow a straightforward process. The first crucial step is to set the function equal to zero. This is because, at the x-intercepts, the y-value, which is represented by f(x), is zero. So, we have the equation (x-4)(x+2) = 0. This equation is already factored, which makes our task significantly easier. If the quadratic function was not in factored form, we would need to factor it first or use the quadratic formula to find the roots. However, since we have the factored form, we can proceed directly to the next step. The second step involves applying the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, this means either (x-4) = 0 or (x+2) = 0. The final step is to solve each of these linear equations for x. Solving x - 4 = 0 gives us x = 4, and solving x + 2 = 0 gives us x = -2. These values of x are the x-coordinates of the x-intercepts. Therefore, the x-intercepts are the points where the graph of the function intersects the x-axis, and we have found them by setting f(x) to zero and solving for x. This method is a fundamental technique in algebra and is widely used to find the roots of various types of functions.

Solving the Equation (x-4)(x+2) = 0

Now, let's delve deeper into the process of solving the equation (x-4)(x+2) = 0. As mentioned earlier, this equation is already conveniently factored for us, which simplifies the process of finding the x-intercepts. The key principle we employ here is the zero-product property. This property is a cornerstone of algebra and is particularly useful when dealing with factored equations. It states that if the product of two or more factors is equal to zero, then at least one of those factors must be equal to zero. Applying this to our equation, we can deduce that either the factor (x-4) must be equal to zero, or the factor (x+2) must be equal to zero, or both. This gives us two separate linear equations to solve: x - 4 = 0 and x + 2 = 0. To solve the first equation, x - 4 = 0, we simply add 4 to both sides of the equation. This isolates x on the left side and gives us x = 4. Similarly, to solve the second equation, x + 2 = 0, we subtract 2 from both sides of the equation. This again isolates x and gives us x = -2. These two solutions, x = 4 and x = -2, represent the x-coordinates of the x-intercepts of the quadratic function. They are the points where the parabola intersects the x-axis. This method of using the zero-product property to solve factored equations is a fundamental skill in algebra and is essential for solving a wide range of problems involving polynomials and other types of functions.

Identifying the X-Intercept Points

Having found the x-values where the function intersects the x-axis, we now need to express these as coordinate points. Remember, x-intercepts are points on the coordinate plane, and they are represented as (x, y) coordinates. Since the x-intercepts occur where f(x) = 0, the y-coordinate at these points is always 0. We found that the solutions to the equation (x-4)(x+2) = 0 are x = 4 and x = -2. Therefore, the x-intercepts are the points where x is 4 and -2, and y is 0. This gives us the x-intercept points (4, 0) and (-2, 0). These points are crucial for understanding the behavior of the quadratic function. They tell us where the parabola crosses the x-axis, and they can also help us determine the axis of symmetry and the vertex of the parabola. By plotting these points on a graph, we can visualize the parabola and its relationship to the x-axis. The x-intercepts are also important in real-world applications. For example, if this quadratic function represents the path of a projectile, the x-intercepts would represent the points where the projectile hits the ground. Therefore, correctly identifying and interpreting x-intercepts is a fundamental skill in algebra and has significant practical applications.

The Correct Answer

Based on our calculations, the x-intercepts of the quadratic function f(x) = (x-4)(x+2) are the points (4, 0) and (-2, 0). Now, let's consider the options provided in the original question:

A. (-4, 0) B. (-2, 0) C. (0, 2) D. (4, -2)

Comparing our results with the given options, we can see that option B, (-2, 0), and a modified version of option A, but with the correct sign, which would be * (4,0)* match the x-intercepts we calculated. Option C, (0, 2), represents the y-intercept (where the graph crosses the y-axis), not an x-intercept. Option D, (4, -2), is not an x-intercept because the y-coordinate is not zero. Therefore, the correct answer is B. (-2, 0) and a modified A. (4,0). This exercise highlights the importance of carefully following the steps to find x-intercepts and accurately interpreting the results. By setting f(x) = 0 and solving for x, we can confidently identify the points where the graph of the function intersects the x-axis.

Conclusion

In conclusion, finding the x-intercepts of a quadratic function is a fundamental skill in algebra. For the function f(x) = (x-4)(x+2), we identified the x-intercepts by setting the function equal to zero and solving for x. This process involved applying the zero-product property, which allowed us to find the x-values where the graph intersects the x-axis. We determined that the x-intercepts are the points (-2, 0) and (4, 0). Understanding how to find x-intercepts is crucial for analyzing the behavior of quadratic functions and their graphs, and it has practical applications in various fields. By mastering this skill, you'll be well-equipped to tackle more complex algebraic problems and gain a deeper understanding of mathematical functions. For further exploration of quadratic functions and their properties, consider visiting Khan Academy's Algebra section.