Finding The Vertex Of G(x) = 8x² - 64x A Comprehensive Guide

Introduction: Understanding Quadratic Functions and Their Vertices

In the realm of mathematics, quadratic functions hold a prominent position, particularly in algebra and calculus. These functions, characterized by their parabolic curves, find applications in diverse fields, ranging from physics and engineering to economics and computer science. Among the key features of a parabola, the vertex stands out as a crucial point, representing either the maximum or minimum value of the function. In this article, we delve into the intricacies of quadratic functions, focusing on how to determine the vertex. Our specific example will be the function g(x) = 8x² - 64x, a classic illustration of a quadratic equation. We'll explore different methods to find the vertex, ensuring a comprehensive understanding of this fundamental concept.

Before we plunge into the specifics of g(x) = 8x² - 64x, let's establish a firm grasp on the general form of a quadratic function. A quadratic function is typically expressed as f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The coefficient a dictates the parabola's direction: if a is positive, the parabola opens upwards, indicating a minimum vertex; if a is negative, it opens downwards, indicating a maximum vertex. The vertex, in essence, is the turning point of the parabola, a critical feature for understanding the function's behavior. Knowing how to find the vertex not only provides valuable information about the function's range and extreme values but also aids in graphing the parabola accurately. This understanding is crucial for solving optimization problems, predicting trajectories, and analyzing various real-world scenarios modeled by quadratic functions. So, let's embark on this journey to master the art of finding the vertex, starting with our example function, g(x) = 8x² - 64x.

Method 1: Completing the Square – A Step-by-Step Approach

One of the most effective methods for finding the vertex of a quadratic function is completing the square. This technique transforms the standard form of the quadratic equation, f(x) = ax² + bx + c, into vertex form, f(x) = a(x - h)² + k, where (h, k) represents the vertex coordinates. For our function, g(x) = 8x² - 64x, we will systematically apply this method to unveil the vertex. The initial step involves factoring out the coefficient of the x² term from the first two terms of the equation. In this case, we factor out 8 from 8x² - 64x, resulting in 8(x² - 8x). This step sets the stage for creating a perfect square trinomial within the parentheses.

Next, we focus on the expression inside the parentheses, x² - 8x. To complete the square, we take half of the coefficient of the x term, which is -8, and square it. Half of -8 is -4, and squaring -4 gives us 16. We then add and subtract this value inside the parentheses to maintain the equation's balance. This gives us 8(x² - 8x + 16 - 16). Notice that x² - 8x + 16 is a perfect square trinomial, which can be factored into (x - 4)². Now, our equation looks like 8((x - 4)² - 16). The final step is to distribute the 8 back into the parentheses. This yields 8(x - 4)² - 8 * 16, which simplifies to 8(x - 4)² - 128. We have now successfully converted our function into vertex form: g(x) = 8(x - 4)² - 128. By comparing this to the vertex form f(x) = a(x - h)² + k, we can easily identify the vertex coordinates. The h value is 4, and the k value is -128. Therefore, the vertex of the function g(x) = 8x² - 64x is (4, -128). This meticulous step-by-step approach of completing the square not only reveals the vertex but also enhances our understanding of the function's structure and behavior.

Method 2: The Vertex Formula – A Direct Approach

Another efficient method for finding the vertex of a quadratic function is using the vertex formula. This formula provides a direct way to calculate the coordinates of the vertex without the need for completing the square. For a quadratic function in the standard form f(x) = ax² + bx + c, the x-coordinate of the vertex, denoted as h, is given by the formula h = -b / 2a. Once we find h, we can substitute it back into the original function to find the y-coordinate of the vertex, denoted as k. This method is particularly useful when a quick and straightforward solution is desired.

Applying the vertex formula to our function, g(x) = 8x² - 64x, we first identify the coefficients a and b. In this case, a = 8 and b = -64. Using the formula h = -b / 2a, we substitute these values to find the x-coordinate of the vertex: h = -(-64) / (2 * 8) = 64 / 16 = 4. Thus, the x-coordinate of the vertex is 4. To find the y-coordinate, k, we substitute h = 4 back into the function g(x): g(4) = 8(4)² - 64(4) = 8(16) - 256 = 128 - 256 = -128. Therefore, the y-coordinate of the vertex is -128. Combining these results, we find that the vertex of the function g(x) = 8x² - 64x is (4, -128), which aligns perfectly with our findings from the completing the square method. The vertex formula offers a concise and effective way to determine the vertex, especially when dealing with more complex quadratic functions. It serves as a valuable tool in our mathematical arsenal for analyzing parabolas and their properties.

Method 3: Utilizing Symmetry – A Graphical Perspective

Quadratic functions, with their parabolic nature, possess a unique characteristic: symmetry. The parabola is symmetrical about a vertical line that passes through its vertex, known as the axis of symmetry. This symmetry provides another avenue for determining the vertex. The x-coordinate of the vertex lies exactly in the middle of the roots (or x-intercepts) of the quadratic function. Therefore, if we can find the roots of the function, we can easily calculate the x-coordinate of the vertex by averaging them. This method offers a graphical perspective on finding the vertex and can be particularly insightful when visualizing the parabola.

To apply this method to our function, g(x) = 8x² - 64x, we first need to find the roots. The roots are the values of x for which g(x) = 0. So, we set 8x² - 64x = 0 and solve for x. We can factor out an 8x from the equation, giving us 8x(x - 8) = 0. This equation is satisfied when either 8x = 0 or x - 8 = 0. Solving these equations, we find the roots to be x = 0 and x = 8. Now, to find the x-coordinate of the vertex, we average these roots: (0 + 8) / 2 = 4. Thus, the x-coordinate of the vertex is 4. To find the y-coordinate, we substitute x = 4 back into the function g(x), as we did in the vertex formula method: g(4) = 8(4)² - 64(4) = 8(16) - 256 = 128 - 256 = -128. This gives us the y-coordinate of the vertex as -128. Therefore, the vertex of the function g(x) = 8x² - 64x, using the symmetry method, is (4, -128). This method not only reinforces our understanding of the vertex but also highlights the symmetrical nature of parabolas and the relationship between roots and the vertex.

Conclusion: Solidifying Our Understanding of the Vertex

In this comprehensive exploration, we have successfully determined the vertex of the quadratic function g(x) = 8x² - 64x using three distinct methods: completing the square, the vertex formula, and utilizing symmetry. Each method provided a unique perspective and reinforced the fundamental concept of the vertex as the turning point of the parabola. Through completing the square, we transformed the function into vertex form, directly revealing the vertex coordinates. The vertex formula offered a quick and efficient way to calculate the vertex using the coefficients of the quadratic equation. Lastly, leveraging the symmetry of the parabola, we found the vertex by averaging the roots of the function. All three methods converged on the same result: the vertex of g(x) = 8x² - 64x is (4, -128).

This exercise not only demonstrates the versatility of mathematical techniques but also underscores the importance of understanding quadratic functions and their properties. The ability to find the vertex is crucial in various applications, from optimizing processes to modeling physical phenomena. Mastering these methods equips us with the tools to analyze and interpret quadratic functions effectively. As we conclude this journey, it's clear that a solid grasp of the vertex, and the methods to find it, is an invaluable asset in the realm of mathematics and its diverse applications.