Finding The Vertex Of The Parabola Y=(x-1)^2-5 A Step-by-Step Guide

Introduction: Understanding the Vertex

In the realm of quadratic functions, the vertex holds a position of paramount importance. It represents the point where the parabola, the graphical representation of a quadratic equation, changes direction. Identifying the vertex is crucial for understanding the behavior of the function, including its minimum or maximum value and its axis of symmetry. This article will delve into the process of determining the vertex of a parabola, specifically focusing on the equation y = (x - 1)² - 5. We will explore the standard form of a quadratic equation, the vertex form, and how to transform an equation into vertex form to easily identify the vertex. Understanding the vertex is not just a mathematical exercise; it has practical applications in various fields, such as physics (projectile motion), engineering (designing parabolic reflectors), and economics (modeling cost curves). The vertex provides critical information about the function's behavior and allows us to make predictions and optimizations based on its properties.

To truly grasp the concept of the vertex, it's essential to first understand the fundamental characteristics of parabolas. A parabola is a symmetrical, U-shaped curve defined by a quadratic equation of the form y = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The symmetry of the parabola is centered around its axis of symmetry, a vertical line that passes through the vertex. The x-coordinate of the vertex corresponds to the equation of the axis of symmetry. The y-coordinate of the vertex represents either the minimum value (if the parabola opens upwards, i.e., 'a' is positive) or the maximum value (if the parabola opens downwards, i.e., 'a' is negative) of the function. Therefore, determining the vertex provides valuable insights into the extreme values of the quadratic function, which is crucial in many real-world applications. For instance, in optimization problems, identifying the vertex helps find the maximum profit, minimum cost, or the optimal trajectory of a projectile. The vertex also plays a key role in graphing parabolas, as it serves as the anchor point around which the rest of the curve is drawn. By understanding the vertex and its relationship to the parabola's equation, we can effectively analyze and interpret quadratic functions in various contexts.

The equation y = (x - 1)² - 5 is presented in a special form known as the vertex form of a quadratic equation. This form, y = a(x - h)² + k, is incredibly useful because it directly reveals the coordinates of the vertex. In this form, (h, k) represents the vertex of the parabola. Comparing the given equation to the vertex form, we can readily identify the values of h and k. This is a significant advantage over the standard form (y = ax² + bx + c), where the vertex coordinates are not immediately apparent. To find the vertex from the standard form, one typically needs to complete the square or use the formula x = -b / 2a to find the x-coordinate of the vertex, and then substitute this value back into the equation to find the y-coordinate. However, with the vertex form, these steps are bypassed, making it a more efficient and intuitive way to determine the vertex. The value of 'a' in the vertex form still plays a crucial role; it determines the direction in which the parabola opens and the steepness of the curve. A positive 'a' indicates an upward-opening parabola, while a negative 'a' indicates a downward-opening parabola. The larger the absolute value of 'a', the steeper the parabola. Therefore, understanding the vertex form not only simplifies the process of finding the vertex but also provides a deeper understanding of the parabola's shape and orientation. This makes it a valuable tool in analyzing and graphing quadratic functions.

Identifying the Vertex of y=(x-1)^2-5

To pinpoint the vertex of the graph represented by the equation y = (x - 1)² - 5, we leverage the vertex form of a quadratic equation. The vertex form, as previously mentioned, is expressed as y = a(x - h)² + k, where the coordinates of the vertex are given by (h, k). By carefully comparing our given equation y = (x - 1)² - 5 with the vertex form, we can directly extract the values of h and k. In this case, we can see that a = 1, h = 1, and k = -5. Therefore, the vertex of the parabola is located at the point (1, -5). This method provides a straightforward and efficient way to determine the vertex without the need for complex calculations or manipulations. The vertex form essentially encodes the vertex coordinates within the equation itself, making it a powerful tool for analyzing quadratic functions.

The beauty of the vertex form lies in its ability to directly reveal the vertex coordinates. By simply recognizing the pattern and extracting the values of h and k, we bypass the need for completing the square or using the formula x = -b / 2a. This direct identification method is particularly useful when dealing with equations already presented in vertex form or when the primary goal is to quickly determine the vertex. The vertex, being the turning point of the parabola, holds significant information about the function's behavior. It indicates the minimum or maximum value of the function and serves as the point of symmetry for the parabola. The x-coordinate of the vertex represents the axis of symmetry, while the y-coordinate represents the minimum or maximum value of the function. In the case of y = (x - 1)² - 5, since the coefficient of the squared term (a) is positive (1), the parabola opens upwards, and the vertex (1, -5) represents the minimum point of the function. This means that the function's lowest value is -5, which occurs when x = 1. Understanding the significance of the vertex allows us to quickly analyze and interpret the behavior of the quadratic function and its graph.

Furthermore, identifying the vertex is crucial for graphing the parabola. The vertex serves as the central point around which the rest of the parabola is drawn. Once the vertex is located, we can use the value of 'a' to determine the direction and steepness of the parabola. In the case of y = (x - 1)² - 5, we know that the parabola opens upwards (since a = 1) and has a standard width (since a = 1). We can then find additional points on the parabola by substituting different values of x into the equation and calculating the corresponding y values. The symmetry of the parabola ensures that for every point (x, y) on one side of the axis of symmetry, there is a corresponding point (2h - x, y) on the other side. This symmetry simplifies the process of graphing, as we only need to calculate points on one side of the vertex. By plotting the vertex and a few additional points, we can accurately sketch the parabola and visualize the behavior of the quadratic function. This graphical representation provides a powerful tool for understanding the function's properties, including its domain, range, intercepts, and the relationship between input and output values. Therefore, identifying the vertex is a fundamental step in analyzing and visualizing quadratic functions.

The Vertex: (1, -5)

Based on our analysis, the vertex of the graph of the equation y = (x - 1)² - 5 is indeed the point (1, -5). This signifies that the parabola reaches its minimum value at x = 1, where the value of y is -5. This point serves as the turning point of the parabola, where it changes direction from decreasing to increasing. The vertex is a critical feature of the parabola, providing insights into its overall shape, position, and behavior. Its coordinates directly influence the graph's symmetry and extreme values, making it a fundamental element in understanding quadratic functions. The ease with which we identified the vertex using the vertex form highlights the importance of this form in analyzing quadratic equations.

This vertex (1, -5) also dictates the axis of symmetry for the parabola. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation of the axis of symmetry is given by x = h, where h is the x-coordinate of the vertex. In this case, the axis of symmetry is the vertical line x = 1. This means that for any two points on the parabola with the same y-coordinate, their x-coordinates will be equidistant from the line x = 1. This symmetry simplifies the process of graphing the parabola, as we only need to find points on one side of the axis of symmetry and then reflect them across the line to obtain corresponding points on the other side. The axis of symmetry is a direct consequence of the quadratic function's symmetrical nature and is a valuable tool for understanding and visualizing the parabola's behavior. The vertex, therefore, not only provides the minimum or maximum value of the function but also defines the line of symmetry around which the parabola is balanced.

In summary, the vertex (1, -5) provides a wealth of information about the quadratic function y = (x - 1)² - 5. It represents the minimum point of the parabola, dictates the axis of symmetry, and serves as the central point for graphing the function. Its easy identification through the vertex form underscores the importance of this form in analyzing quadratic equations. Understanding the vertex allows us to quickly grasp the key characteristics of the parabola, including its shape, position, and extreme values. This knowledge is crucial in various applications, from solving optimization problems to modeling real-world phenomena. The vertex, therefore, is not just a point on the graph; it is a fundamental concept that unlocks a deeper understanding of quadratic functions and their applications.

Conclusion

In conclusion, we have successfully identified the vertex of the graph of y = (x - 1)² - 5 as the point (1, -5). This was achieved by recognizing and applying the vertex form of a quadratic equation, which provides a direct and efficient method for determining the vertex coordinates. The vertex, as the turning point of the parabola, holds significant information about the function's behavior, including its minimum value and axis of symmetry. Understanding the vertex form and its relationship to the parabola's characteristics is crucial for analyzing and graphing quadratic functions. This knowledge has far-reaching applications in various fields, making the concept of the vertex a fundamental tool in mathematics and beyond.