Identifying Functions With A Vertex At The Origin

In the realm of mathematics, understanding the characteristics of different functions is crucial. One key feature of a quadratic function is its vertex, which represents the minimum or maximum point on the parabola. Identifying the function with a vertex at the origin, the point (0, 0), requires careful analysis of its equation. This article delves into the process of determining which function among a given set has its vertex at the origin, providing a comprehensive guide with explanations and examples. We will explore the standard forms of quadratic equations and how they relate to the vertex's location. Understanding the vertex form of a quadratic equation, f(x) = a(x - h)² + k, is crucial, as the vertex is directly given by the point (h, k). The goal is to identify which function, when expressed in vertex form, will have both h and k equal to zero. This ensures that the parabola's turning point is precisely at the origin. Additionally, we will examine how to convert other forms of quadratic equations, such as the standard form f(x) = ax² + bx + c and the factored form, into vertex form. This conversion process involves completing the square or using the formula h = -b / 2a to find the x-coordinate of the vertex, and then substituting this value back into the original equation to find the y-coordinate. By understanding these transformations, you can easily determine the vertex of any quadratic function and identify those that are situated at the origin. This knowledge is not only fundamental in mathematics but also has practical applications in various fields, including physics, engineering, and computer science. This article aims to provide a clear and thorough understanding of how to identify functions with vertices at the origin, equipping you with the tools necessary to solve related problems and apply this knowledge in real-world scenarios.

Analyzing Quadratic Functions and Their Vertices

To effectively determine which function has a vertex at the origin, it's essential to grasp the concept of a vertex in the context of quadratic functions. Quadratic functions, characterized by their parabolic shape, have a unique turning point known as the vertex. This vertex represents either the minimum or maximum value of the function, depending on whether the parabola opens upwards or downwards. The location of the vertex is a critical attribute of the quadratic function, providing insights into its behavior and graph. To identify a function with a vertex at the origin, we need to analyze its equation and determine the coordinates of its vertex. The vertex form of a quadratic equation is particularly useful for this purpose. As mentioned earlier, the vertex form is given by f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. If a function's equation can be expressed in this form, we can directly read off the vertex coordinates. For instance, if a function is given as f(x) = (x - 2)² + 3, the vertex is located at (2, 3). Conversely, if the vertex is at the origin (0, 0), the equation in vertex form would look like f(x) = ax², where a is a non-zero constant. This is because when h and k are both zero, the equation simplifies to this form. However, quadratic functions are not always presented in vertex form. They can also be given in standard form, f(x) = ax² + bx + c, or factored form, f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the roots of the equation. To find the vertex in these cases, we need to convert the equation into vertex form. This can be achieved through a process called completing the square or by using the formula h = -b / 2a to find the x-coordinate of the vertex. Once we have the x-coordinate, we can substitute it back into the original equation to find the y-coordinate. By mastering these techniques, we can efficiently analyze any quadratic function and determine whether its vertex lies at the origin. This skill is fundamental for solving a wide range of mathematical problems and understanding the behavior of quadratic functions in various applications.

Evaluating the Given Options

Now, let's apply our understanding of quadratic functions and their vertices to the specific options provided. We are tasked with identifying which function has a vertex at the origin. Each option presents a different quadratic function, and we need to analyze each one to determine its vertex coordinates. Option a, f(x) = (x + 4)², is already in a form that closely resembles vertex form. We can rewrite it as f(x) = 1(x - (-4))² + 0. Comparing this to the vertex form f(x) = a(x - h)² + k, we see that h = -4 and k = 0. Therefore, the vertex of this function is at (-4, 0), which is not the origin. This eliminates option a. Option b, f(x) = x(x - 4), is given in factored form. To find the vertex, we first need to expand the equation to standard form: f(x) = x² - 4x. In this form, we can identify a = 1, b = -4, and c = 0. To find the x-coordinate of the vertex, we use the formula h = -b / 2a. Substituting the values, we get h = -(-4) / (2 * 1) = 2. Now, we substitute x = 2 back into the equation to find the y-coordinate: f(2) = 2(2 - 4) = -4. Thus, the vertex of this function is at (2, -4), which is also not the origin. This eliminates option b. Option c, f(x) = (x - 4)(x + 4), is also in factored form. Expanding this equation, we get f(x) = x² - 16. This is a difference of squares, and it can be rewritten in vertex form as f(x) = 1(x - 0)² - 16. Here, h = 0 and k = -16, so the vertex is at (0, -16), which is not the origin. This eliminates option c. Option d, f(x) = -x², is the function we need to examine. This equation can be rewritten in vertex form as f(x) = -1(x - 0)² + 0. In this form, it's clear that h = 0 and k = 0. Therefore, the vertex of this function is at (0, 0), which is the origin. This confirms that option d is the correct answer. Through this step-by-step analysis, we have successfully identified the function with a vertex at the origin.

The Correct Answer: f(x) = -x²

After carefully analyzing each option, we have determined that the function f(x) = -x² has a vertex at the origin (0, 0). This function represents a parabola that opens downwards due to the negative coefficient of the x² term. The vertex form of this function is f(x) = -1(x - 0)² + 0, which clearly shows that the vertex is located at (0, 0). The simplicity of this equation highlights a key characteristic of quadratic functions with vertices at the origin: they often have a straightforward form where the linear and constant terms are absent. This is because the absence of these terms ensures that the parabola's turning point is precisely at the intersection of the x and y axes. The graph of f(x) = -x² is a classic example of a parabola with its vertex at the origin. It is symmetrical about the y-axis, and its highest point is the origin itself. The negative sign in front of the x² term indicates that the parabola opens downwards, meaning the function has a maximum value at the vertex. Understanding this concept is crucial for quickly identifying functions with vertices at the origin. In contrast, the other options presented in the question had vertices located elsewhere. Functions like f(x) = (x + 4)² and f(x) = (x - 4)(x + 4) have vertices that are shifted away from the origin due to the presence of linear or constant terms. The process of analyzing these functions involved either rewriting them in vertex form or using the vertex formula h = -b / 2a to determine the vertex coordinates. By systematically examining each option and comparing their vertex locations, we were able to confidently identify f(x) = -x² as the function with a vertex at the origin. This exercise demonstrates the importance of understanding the different forms of quadratic equations and how they relate to the vertex's position.

Conclusion: Mastering Vertex Identification

In conclusion, identifying the function with a vertex at the origin is a fundamental skill in mathematics, particularly when dealing with quadratic functions. This article has provided a comprehensive guide to this process, starting with an understanding of the vertex concept and its significance in the context of parabolas. We explored the vertex form of a quadratic equation, f(x) = a(x - h)² + k, and how it directly reveals the vertex coordinates (h, k). The ability to convert quadratic equations from standard or factored form into vertex form is crucial for determining the vertex's location. Techniques such as completing the square and using the formula h = -b / 2a were discussed as effective methods for this conversion. By analyzing the given options, we demonstrated a step-by-step approach to identifying the function with a vertex at the origin. Each option was carefully examined, and its vertex coordinates were determined through either direct observation or transformation into vertex form. This process highlighted the importance of understanding the relationship between the equation's form and the parabola's characteristics. The correct answer, f(x) = -x², was identified as the function with a vertex at (0, 0). This function's simplicity underscores a key principle: quadratic functions with vertices at the origin often have a straightforward form without linear or constant terms. Mastering the identification of vertices is not only essential for solving mathematical problems but also has practical applications in various fields. From physics and engineering to computer science and economics, understanding the behavior of quadratic functions and their vertices is crucial for modeling real-world phenomena. This article has equipped you with the knowledge and tools necessary to confidently identify functions with vertices at the origin, enhancing your mathematical skills and problem-solving abilities. By continuing to practice and apply these concepts, you can further solidify your understanding and excel in mathematics.