Which Function Has Two X-intercepts, One At (0,0) And One At (4,0)?

In the realm of mathematics, identifying functions based on their key characteristics such as x-intercepts is a fundamental skill. X-intercepts, also known as roots or zeros, are the points where the graph of a function intersects the x-axis. These points hold significant information about the behavior and structure of the function. This article delves into the process of determining a function that possesses two specific x-intercepts: one at the origin (0,0) and another at (4,0). We will explore how the x-intercepts directly influence the function's equation and analyze different options to pinpoint the correct one.

Understanding x-Intercepts and Function Equations

Before we dive into the specific problem, it's crucial to understand the relationship between x-intercepts and the equation of a function. An x-intercept occurs when the function's value, denoted as f(x), is equal to zero. Mathematically, if a function f(x) has an x-intercept at a point (a, 0), then f(a) = 0. This principle is the cornerstone of finding functions with specific x-intercepts.

When dealing with polynomial functions, particularly quadratic functions (functions of the form f(x) = ax² + bx + c), the x-intercepts are closely related to the factors of the polynomial. If a quadratic function has x-intercepts at x = a and x = b, then the function can be expressed in the factored form as f(x) = k(x - a)(x - b), where k is a constant. This factored form is incredibly useful for constructing functions with desired x-intercepts. The values a and b are the roots or zeros of the quadratic equation, and they represent where the parabola intersects the x-axis. By understanding this relationship, we can efficiently determine the quadratic function that satisfies the given conditions.

The significance of x-intercepts extends beyond simply identifying points on a graph. They provide critical insights into the function's behavior, such as where the function changes sign (from positive to negative or vice versa) and where it reaches its minimum or maximum values. In practical applications, x-intercepts can represent real-world scenarios, such as the break-even point in a business model or the time at which a projectile hits the ground. Thus, a thorough understanding of x-intercepts is essential for both mathematical analysis and practical problem-solving.

Analyzing the Given x-Intercepts: (0,0) and (4,0)

Our objective is to find a function that has x-intercepts at (0,0) and (4,0). This means that when x = 0, f(x) = 0, and when x = 4, f(x) = 0. These two points provide us with the necessary clues to construct the function's equation. The fact that (0,0) is an x-intercept tells us that the function must have a factor of x. This is because when x = 0, the entire term containing x will become zero, thus making f(x) = 0. The other x-intercept at (4,0) indicates that the function must also have a factor of (x - 4). When x = 4, the term (x - 4) becomes zero, again making f(x) = 0.

Considering these two factors, we can construct a function in the form f(x) = kx(x - 4), where k is a constant. This form ensures that the function will have x-intercepts at x = 0 and x = 4, regardless of the value of k (as long as k is not zero). The constant k will affect the vertical stretch or compression of the function's graph but will not change the x-intercepts. This understanding is crucial because it allows us to narrow down the possible functions significantly.

To further illustrate this, let's consider a specific example. If we choose k = 1, the function becomes f(x) = x(x - 4). Expanding this, we get f(x) = x² - 4x. This is a quadratic function, and its graph is a parabola that opens upwards. The x-intercepts are indeed at (0,0) and (4,0), as required. This confirms that our approach of constructing the function based on its factored form is correct. The factored form not only makes it easy to identify the x-intercepts but also provides a clear understanding of how these intercepts influence the function's behavior. This method is a powerful tool in analyzing and constructing functions with specific characteristics.

Evaluating the Given Options

Now, let's analyze the given options in light of our understanding of x-intercepts and function equations:

A. f(x) = x(x - 4) This option directly matches the form we derived, f(x) = kx(x - 4), with k = 1. When x = 0, f(0) = 0(0 - 4) = 0. When x = 4, f(4) = 4(4 - 4) = 4(0) = 0. Thus, this function has x-intercepts at (0,0) and (4,0), making it a strong candidate.

B. f(x) = x(x + 4) This function has an x-intercept at (0,0) because when x = 0, f(0) = 0(0 + 4) = 0. However, the other x-intercept occurs when (x + 4) = 0, which means x = -4. Therefore, the x-intercepts for this function are at (0,0) and (-4,0), not (4,0). This option does not meet our criteria.

C. f(x) = (x - 4)(x - 4) This function can be rewritten as f(x) = (x - 4)². This is a quadratic function with a repeated root at x = 4. This means that the graph of the function touches the x-axis at (4,0) but does not cross it. Therefore, it has only one x-intercept at (4,0), not two x-intercepts at (0,0) and (4,0). This option is incorrect.

D. f(x) = (x + 4)(x + 4) This function can be rewritten as f(x) = (x + 4)². This is a quadratic function with a repeated root at x = -4. Similar to option C, this function has only one x-intercept at (-4,0), not two x-intercepts at (0,0) and (4,0). This option is also incorrect.

Through this evaluation, it becomes clear that option A is the only function that satisfies the condition of having x-intercepts at both (0,0) and (4,0). The systematic analysis of each option, considering the relationship between x-intercepts and the function's equation, allows us to confidently identify the correct answer. The ability to dissect and interpret function behavior based on its roots is a crucial skill in mathematical problem-solving and analysis.

Conclusion: The Function with x-Intercepts at (0,0) and (4,0)

In conclusion, after a thorough analysis of the given options and a deep dive into the relationship between x-intercepts and function equations, we have definitively identified the function that has two x-intercepts, one at (0,0) and one at (4,0). The correct answer is:

A. f(x) = x(x - 4)

This function, in its factored form, clearly demonstrates the presence of x-intercepts at x = 0 and x = 4. The factor x accounts for the x-intercept at the origin (0,0), while the factor (x - 4) accounts for the x-intercept at (4,0). Expanding this function, we obtain f(x) = x² - 4x, which is a quadratic function representing a parabola that intersects the x-axis at the specified points. This exercise underscores the importance of understanding the fundamental connection between a function's roots and its equation.

The process of identifying functions based on their characteristics, such as x-intercepts, is a cornerstone of mathematical analysis. It requires a solid grasp of algebraic principles and the ability to apply these principles systematically. By breaking down the problem into smaller, manageable steps, we were able to construct the function's equation based on the given x-intercepts and then verify our solution by evaluating the provided options. This methodical approach is applicable to a wide range of mathematical problems and is a valuable skill for students and professionals alike.

Moreover, the exploration of this problem highlights the significance of x-intercepts in understanding a function's behavior. These points not only indicate where the function's graph intersects the x-axis but also provide insights into the function's roots, zeros, and potential real-world applications. The ability to quickly identify and interpret x-intercepts is crucial for problem-solving in various fields, including physics, engineering, economics, and computer science. Therefore, a strong foundation in this concept is essential for success in both academic and practical endeavors. The knowledge and skills gained from solving problems like this enhance our mathematical intuition and problem-solving capabilities, paving the way for tackling more complex challenges in the future.