Analyzing The Quadratic Function F(x) = X² - 2x + 1 A Comprehensive Guide

This article delves into the analysis of the quadratic function f(x) = x² - 2x + 1. We will explore how to determine the vertex of the function, identify whether it represents a maximum or minimum point, and learn how to sketch the graph effectively. Understanding these aspects is crucial for comprehending the behavior of quadratic functions and their applications in various fields.

A) Determining the Vertex of the Function

The vertex of a quadratic function is a crucial point that represents either the minimum or maximum value of the function. In the standard form of a quadratic equation, f(x) = ax² + bx + c, the vertex can be found using a specific formula. Let's apply this to our function, f(x) = x² - 2x + 1. Here, a = 1, b = -2, and c = 1. The x-coordinate of the vertex (xv) is given by the formula xv = -b / 2a. Substituting the values, we get xv = -(-2) / (2 * 1) = 2 / 2 = 1. This calculation pinpoints the x-coordinate of our vertex, a critical step in understanding the function's behavior.

To find the y-coordinate of the vertex (yv), we substitute the x-coordinate (xv) back into the original function. Thus, yv = f(1) = (1)² - 2(1) + 1 = 1 - 2 + 1 = 0. Therefore, the vertex of the function f(x) = x² - 2x + 1 is at the point (1, 0). This point is not just any point on the graph; it's the turning point that dictates whether the parabola opens upwards or downwards. The vertex provides a foundational understanding of the function's range and symmetry, key elements in visualizing its graphical representation. Furthermore, accurately determining the vertex is essential for solving optimization problems where maximum or minimum values are sought, making it a cornerstone concept in both theoretical and applied mathematics.

Calculating the Vertex Coordinates

The vertex of a parabola, defined by a quadratic function, is a pivotal point that signifies either the function's minimum or maximum value. To pinpoint this critical point, we employ specific formulas derived from the standard form of a quadratic equation, which is expressed as f(x) = ax² + bx + c. In our given function, f(x) = x² - 2x + 1, we identify the coefficients as a = 1, b = -2, and c = 1. These coefficients are instrumental in determining the vertex's coordinates. The x-coordinate of the vertex, denoted as xv, is calculated using the formula xv = -b / 2a. By substituting the values from our function, we find xv = -(-2) / (2 * 1) = 2 / 2 = 1. This calculation precisely locates the x-coordinate of the vertex, providing the horizontal position of the parabola's turning point.

Next, we determine the y-coordinate of the vertex, denoted as yv, by substituting the calculated x-coordinate back into the original function. This substitution yields yv = f(1) = (1)² - 2(1) + 1 = 1 - 2 + 1 = 0. Consequently, the vertex of the function f(x) = x² - 2x + 1 is definitively located at the point (1, 0). This vertex serves as a cornerstone for understanding the parabola's orientation and range. If the coefficient 'a' is positive, as in our case (a = 1), the parabola opens upwards, and the vertex represents the minimum point of the function. Conversely, if 'a' were negative, the parabola would open downwards, and the vertex would signify the maximum point. Accurately computing the vertex coordinates is paramount for sketching the graph of the quadratic function, enabling us to visualize its behavior and critical characteristics. Moreover, the vertex plays a vital role in various mathematical applications, such as optimization problems, where the goal is to find the maximum or minimum value of a function, making the precise calculation of its coordinates indispensable.

B) Identifying the Vertex as a Maximum or Minimum Point

To determine whether the vertex represents a maximum or minimum point, we examine the coefficient of the x² term (a) in the quadratic function. In the function f(x) = x² - 2x + 1, the coefficient 'a' is 1. Since 'a' is positive (1 > 0), the parabola opens upwards. This upward-opening shape indicates that the vertex is the lowest point on the graph, making it a minimum point. The sign of 'a' is a direct indicator of the parabola's concavity; a positive 'a' means the parabola is concave up, and a negative 'a' means it is concave down. Understanding this relationship is essential for quickly grasping the overall behavior of the quadratic function without needing to plot numerous points. In practical terms, knowing whether the vertex is a minimum or maximum is critical in applications such as optimization problems, where one aims to find the smallest or largest value of a function within a given domain. For instance, in engineering or economics, determining the minimum cost or maximum profit often involves identifying the vertex of a quadratic function.

The Role of the Leading Coefficient

The leading coefficient, denoted as 'a' in the quadratic function f(x) = ax² + bx + c, plays a pivotal role in determining the nature of the vertex and the overall shape of the parabola. Specifically, the sign of 'a' dictates whether the parabola opens upwards or downwards, directly influencing whether the vertex represents a minimum or maximum point. In our function, f(x) = x² - 2x + 1, the leading coefficient 'a' is 1, which is a positive value. This positive sign signifies that the parabola opens upwards, resembling a U-shape. Consequently, the vertex of this parabola represents the minimum point of the function. This is because, as the parabola opens upwards, the vertex is the lowest point on the curve, and all other points on the graph have y-values greater than the y-value of the vertex.

Conversely, if the leading coefficient 'a' were negative, the parabola would open downwards, forming an inverted U-shape. In such cases, the vertex would represent the maximum point of the function. This is because the vertex would be the highest point on the curve, with all other points having y-values less than the y-value of the vertex. The magnitude of 'a' also influences the parabola's shape; a larger absolute value of 'a' results in a narrower parabola, while a smaller absolute value leads to a wider parabola. Therefore, by simply examining the sign and magnitude of the leading coefficient, we can quickly ascertain whether the vertex is a minimum or maximum point and gain insights into the parabola's overall appearance. This understanding is invaluable in various applications, including optimization problems, physics, and engineering, where the behavior of quadratic functions is often used to model real-world phenomena. For example, in projectile motion, the maximum height of a projectile can be determined by finding the vertex of a downward-opening parabola representing its trajectory.

C) Sketching the Graph

Sketching the graph of the quadratic function f(x) = x² - 2x + 1 involves plotting key points and understanding the shape of the parabola. We already know the vertex is at (1, 0), which is the minimum point. To get a better sense of the graph, we can find additional points by substituting different x-values into the function. For example, when x = 0, f(0) = (0)² - 2(0) + 1 = 1, giving us the point (0, 1). Similarly, when x = 2, f(2) = (2)² - 2(2) + 1 = 1, resulting in the point (2, 1). These points, along with the vertex, provide a basic framework for the parabola.

Plotting Key Points and the Parabola Shape

To accurately sketch the graph of the quadratic function f(x) = x² - 2x + 1, we begin by plotting the key points we have already determined. The vertex, located at (1, 0), is a crucial reference point as it represents the minimum value of the function and the axis of symmetry for the parabola. We also have two additional points, (0, 1) and (2, 1), which provide further guidance on the curve's shape and direction. By plotting these points on a coordinate plane, we can start to visualize the parabolic form of the graph. Since the coefficient 'a' in our function is positive (a = 1), we know that the parabola opens upwards, forming a U-shape with the vertex as its lowest point.

To enhance the accuracy of our sketch, we can consider the symmetry of the parabola. Parabolas are symmetrical about the vertical line that passes through their vertex, known as the axis of symmetry. In this case, the axis of symmetry is the line x = 1. This symmetry means that for any point (x, y) on the parabola, there is a corresponding point (2 - x, y) on the other side of the axis of symmetry. Utilizing this property, we can easily find additional points to refine our sketch. For instance, if we find the point where x = -1, we can calculate f(-1) = (-1)² - 2(-1) + 1 = 1 + 2 + 1 = 4, giving us the point (-1, 4). Due to symmetry, there will be a corresponding point at x = 3, which is (3, 4). By plotting these additional points, we can draw a smoother and more precise curve, accurately representing the graph of the quadratic function f(x) = x² - 2x + 1. Furthermore, understanding the symmetry and the influence of the leading coefficient 'a' allows us to quickly sketch the graph of any quadratic function, making it an essential skill in mathematical analysis and problem-solving. This graphical representation provides a visual understanding of the function's behavior, including its roots, vertex, and direction of opening, which are critical in various applications such as optimization, physics, and engineering.

The parabola's shape is determined by the positive coefficient of x², indicating it opens upwards. The axis of symmetry is the vertical line passing through the vertex, which is x = 1. This means the graph is symmetrical around this line. With the vertex and a few additional points, we can sketch a smooth, U-shaped curve that accurately represents the function f(x) = x² - 2x + 1. This graphical representation provides a visual understanding of the function's behavior, including its roots and minimum value.

Finding Additional Points for Accuracy

To enhance the accuracy of the graph sketch for the quadratic function f(x) = x² - 2x + 1, finding additional points beyond the vertex is crucial. The vertex, located at (1, 0), provides the minimum point of the parabola, but to fully capture the curve's shape, we need more coordinates. We can achieve this by selecting various x-values and substituting them into the function to calculate the corresponding y-values. For example, let's consider x = -1. Substituting this into the function, we get f(-1) = (-1)² - 2(-1) + 1 = 1 + 2 + 1 = 4. This gives us the point (-1, 4) on the graph. Similarly, if we choose x = 3, we find f(3) = (3)² - 2(3) + 1 = 9 - 6 + 1 = 4, resulting in the point (3, 4).

These additional points not only help in drawing a more precise curve but also demonstrate the symmetry of the parabola. Parabolas are symmetrical about their axis of symmetry, which is the vertical line that passes through the vertex. In our case, the axis of symmetry is the line x = 1. This symmetry means that for any point (x, y) on one side of the axis, there is a corresponding point on the other side with the same y-value. By finding points equidistant from the axis of symmetry, we can ensure that our sketch accurately reflects this symmetry. For instance, the points (-1, 4) and (3, 4) are both two units away from the axis of symmetry (x = 1) and have the same y-value, illustrating this principle. Furthermore, calculating points closer to the vertex provides a more detailed view of the curve's behavior near its minimum. By carefully selecting and plotting additional points, we can create a comprehensive and accurate graph of the quadratic function f(x) = x² - 2x + 1, allowing for a deeper understanding of its properties and behavior. This graphical representation is invaluable in various mathematical and real-world applications, providing a visual tool for analyzing the function's characteristics and solving related problems.

By following these steps, you can effectively analyze the quadratic function f(x) = x² - 2x + 1, determine its vertex, identify whether it's a maximum or minimum point, and sketch its graph. This comprehensive approach provides a solid foundation for understanding quadratic functions and their applications.