Determining The Vertex Of F(x) = 2(x - 4)^2 + 5 A Comprehensive Guide

In mathematics, understanding the properties of functions is crucial, especially when dealing with quadratic functions. Quadratic functions, characterized by their parabolic shape, are prevalent in various fields, from physics to engineering. A key feature of a parabola is its vertex, which represents the minimum or maximum point of the function. Determining the vertex is essential for solving optimization problems, graphing functions, and understanding the behavior of quadratic equations. This article delves into a step-by-step approach to identify the vertex of a quadratic function, using the example function f(x) = 2(x - 4)^2 + 5 as a practical illustration. We will explore the standard form of a quadratic function, the vertex form, and how to extract the vertex coordinates directly from the equation. Understanding these concepts will empower you to analyze and manipulate quadratic functions effectively.

Before diving into the specifics of finding the vertex, it's essential to grasp the fundamental concepts of quadratic functions. A quadratic function is generally expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards, depending on the sign of the coefficient a. If a > 0, the parabola opens upwards, indicating a minimum point (the vertex). Conversely, if a < 0, the parabola opens downwards, indicating a maximum point (the vertex). The vertex is the turning point of the parabola and plays a significant role in understanding the function's behavior. The vertex represents the extreme value of the function – the lowest point if the parabola opens upwards and the highest point if it opens downwards. In addition to the standard form, quadratic functions can also be expressed in vertex form, which provides direct information about the vertex coordinates. Understanding the relationship between the coefficients and the shape of the parabola is crucial for effectively analyzing quadratic functions. This foundational knowledge sets the stage for accurately determining the vertex and utilizing it for various mathematical applications. By recognizing the key characteristics of quadratic functions, such as the direction of opening and the significance of the vertex, you can gain deeper insights into their behavior and solve related problems with greater confidence.

The vertex form of a quadratic function is a powerful tool for easily identifying the vertex of a parabola. The vertex form is given by f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. This form provides a direct way to read off the vertex without any additional calculations. The constant a in the vertex form is the same as the a in the standard form (f(x) = ax^2 + bx + c) and determines the direction and steepness of the parabola. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards. The absolute value of a indicates how stretched or compressed the parabola is compared to the basic parabola f(x) = x^2. The value of h represents the horizontal shift of the parabola from the origin, and k represents the vertical shift. Specifically, if h is positive, the parabola is shifted to the right by h units, and if h is negative, it's shifted to the left by |h| units. Similarly, if k is positive, the parabola is shifted upwards by k units, and if k is negative, it's shifted downwards by |k| units. The vertex (h, k) is the point where the parabola changes direction, making it a critical feature for understanding the function's behavior. By expressing a quadratic function in vertex form, you can immediately determine the vertex and gain valuable insights into the parabola's position and orientation in the coordinate plane. This form simplifies the process of graphing quadratic functions and solving optimization problems.

Now, let's apply the concept of vertex form to the given function, f(x) = 2(x - 4)^2 + 5. By comparing this function to the general vertex form f(x) = a(x - h)^2 + k, we can directly identify the values of a, h, and k. In this case, we have a = 2, h = 4, and k = 5. The value of a = 2 indicates that the parabola opens upwards (since a > 0) and is vertically stretched compared to the basic parabola f(x) = x^2. The values of h and k provide the coordinates of the vertex. Since h = 4, the parabola is shifted 4 units to the right from the origin. Since k = 5, the parabola is shifted 5 units upwards from the origin. Therefore, the vertex of the parabola is the point (h, k) = (4, 5). This means that the minimum value of the function occurs at x = 4, and the minimum value is f(4) = 5. The vertex form of the function clearly reveals the location of the vertex, making it easy to understand the parabola's position in the coordinate plane. By recognizing the values of h and k, we can quickly determine the vertex without needing to complete the square or use other methods. This direct approach is one of the main advantages of using the vertex form of a quadratic function. Understanding how to extract the vertex coordinates from the vertex form is essential for analyzing and graphing quadratic functions efficiently.

To determine the vertex of the function f(x) = 2(x - 4)^2 + 5, we follow a straightforward step-by-step process using the vertex form. First, identify the vertex form of a quadratic function, which is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex. Second, compare the given function f(x) = 2(x - 4)^2 + 5 with the vertex form. By doing so, we can directly extract the values of a, h, and k. In this case, it's clear that a = 2, h = 4, and k = 5. Third, recognize that the vertex coordinates are given by (h, k). Since we have identified h = 4 and k = 5, the vertex of the function is the point (4, 5). This means that the x-coordinate of the vertex is 4, and the y-coordinate is 5. The vertex represents the minimum point of the parabola because a > 0, indicating that the parabola opens upwards. Finally, confirm your result by understanding that the vertex is the turning point of the parabola, and its coordinates directly correspond to the values of h and k in the vertex form. This step-by-step method provides a clear and concise way to determine the vertex of a quadratic function when it is given in vertex form. By following these steps, you can easily identify the vertex and gain insights into the function's behavior and graph. Understanding the relationship between the vertex form and the vertex coordinates is crucial for effectively analyzing quadratic functions.

In conclusion, by analyzing the function f(x) = 2(x - 4)^2 + 5, we have successfully determined the vertex of the parabola. By recognizing the function's vertex form, f(x) = a(x - h)^2 + k, and comparing it to the given function, we identified h = 4 and k = 5. Therefore, the vertex of the function is the point (4, 5). This result signifies that the parabola has a minimum point at x = 4, with a corresponding function value of 5. The vertex is a crucial characteristic of a quadratic function, as it represents the turning point and provides valuable information about the function's behavior and graph. Understanding how to determine the vertex is essential for solving optimization problems, sketching parabolas, and analyzing quadratic equations. The vertex form of a quadratic function provides a direct and efficient method for identifying the vertex coordinates, making it a fundamental tool in mathematical analysis. By mastering this concept, you can confidently analyze and manipulate quadratic functions in various contexts. The process of determining the vertex not only helps in understanding the specific function but also enhances your overall mathematical problem-solving skills. The ability to recognize and utilize the vertex form is a key step in mastering quadratic functions and their applications.