Finding The Vertex Of G(x) = |x-8| + 6 A Step-by-Step Guide

by THE IDEN 60 views

Introduction to Absolute Value Functions and Their Graphs

In the fascinating world of mathematics, absolute value functions hold a special place, characterized by their distinctive V-shaped graphs. Understanding these functions is crucial for grasping various mathematical concepts, from solving equations and inequalities to analyzing transformations of graphs. In this comprehensive exploration, we will delve into the specifics of the absolute value function g(x) = |x-8| + 6, with a primary focus on identifying its vertex. The vertex, in this context, represents the point where the graph changes direction, marking a crucial feature of the function's behavior. Identifying the vertex not only helps in visualizing the graph but also provides valuable insights into the function's minimum or maximum value. We will embark on a step-by-step journey to dissect this function, understand its components, and ultimately pinpoint the coordinates of its vertex. Our exploration will involve a blend of graphical and algebraic techniques, ensuring a thorough understanding of the underlying principles. By the end of this analysis, you will not only be able to determine the vertex of this specific function but also gain a broader understanding of how to approach similar problems involving absolute value functions. This knowledge will empower you to tackle a range of mathematical challenges with confidence and clarity. So, let's begin our mathematical adventure and unravel the mystery of the vertex.

Understanding the Components of g(x) = |x-8| + 6

To effectively determine the vertex of the function g(x) = |x-8| + 6, it is essential to first dissect its components and understand how each part contributes to the overall shape and position of the graph. At its core, this function is an absolute value function, denoted by the vertical bars | |. The absolute value of a number is its distance from zero, always resulting in a non-negative value. This characteristic is what gives the absolute value function its distinctive V-shape. Within the absolute value expression, we have x - 8. This term represents a horizontal shift of the basic absolute value function |x|. Specifically, subtracting 8 from x causes the graph to shift 8 units to the right. This shift is a crucial aspect in determining the vertex's x-coordinate. The addition of + 6 outside the absolute value expression signifies a vertical shift. Adding 6 to the entire expression moves the graph 6 units upwards. This vertical shift directly impacts the y-coordinate of the vertex. By recognizing these transformations, we can start to visualize how the basic absolute value function |x| is modified to become g(x) = |x-8| + 6. Understanding these shifts is key to accurately locating the vertex, which is the point where the V-shape changes direction. The vertex represents the minimum value of the function, and its coordinates provide valuable information about the function's behavior and graph.

Method 1: Graphical Approach to Finding the Vertex

One of the most intuitive ways to find the vertex of the absolute value function g(x) = |x-8| + 6 is through a graphical approach. This method involves visualizing the function's graph and identifying the point where it changes direction, which is the vertex. To begin, let's consider the basic absolute value function f(x) = |x|. Its graph is a V-shaped curve with the vertex at the origin (0, 0). Now, let's analyze how the transformations in g(x) = |x-8| + 6 affect this basic graph. As we discussed earlier, the x - 8 term inside the absolute value shifts the graph 8 units to the right along the x-axis. This means the vertex of |x| moves from (0, 0) to (8, 0). Next, the + 6 term outside the absolute value shifts the graph 6 units upwards along the y-axis. This vertical shift moves the vertex from (8, 0) to (8, 6). Therefore, by understanding these transformations, we can visually trace the movement of the vertex from the basic function to the transformed function. To further solidify this understanding, you can plot the graph of g(x) = |x-8| + 6 using graphing software or by manually plotting points. The graph will clearly show the V-shape with its lowest point at (8, 6), confirming our graphical analysis. This graphical approach provides a visual and intuitive way to understand how transformations affect the vertex of an absolute value function.

Method 2: Algebraic Approach to Determining the Vertex

While the graphical approach provides a visual understanding of the vertex, an algebraic approach offers a more precise and analytical method for determining its coordinates. This method relies on understanding the properties of absolute value functions and applying algebraic principles. The key to finding the vertex algebraically lies in recognizing that the minimum value of the absolute value expression |x-8| is 0. This occurs when the expression inside the absolute value, x-8, is equal to 0. Solving the equation x - 8 = 0, we find that x = 8. This value of x corresponds to the x-coordinate of the vertex. Now, to find the y-coordinate of the vertex, we substitute this value of x back into the original function g(x) = |x-8| + 6. So, g(8) = |8-8| + 6 = |0| + 6 = 0 + 6 = 6. Therefore, the y-coordinate of the vertex is 6. Combining the x-coordinate and y-coordinate, we find that the vertex of the function g(x) = |x-8| + 6 is (8, 6). This algebraic method provides a direct and accurate way to determine the vertex without relying on graphical representations. It demonstrates the power of algebraic manipulation in understanding the behavior of functions. By setting the expression inside the absolute value to zero, we effectively find the point where the function reaches its minimum value, which is the vertex.

Comparing Graphical and Algebraic Methods

Both the graphical and algebraic methods offer valuable insights into finding the vertex of the absolute value function g(x) = |x-8| + 6. However, they approach the problem from different perspectives and offer unique advantages. The graphical method provides a visual understanding of how transformations affect the function's graph. By tracing the shifts and stretches, one can intuitively grasp the movement of the vertex. This method is particularly helpful for visualizing the overall behavior of the function and understanding the impact of different parameters. On the other hand, the algebraic method provides a precise and analytical solution. By setting the expression inside the absolute value to zero and solving for x, we can directly calculate the x-coordinate of the vertex. Substituting this value back into the function gives us the y-coordinate. This method is less prone to errors that might arise from visual estimations in the graphical approach. In terms of efficiency, the algebraic method is generally faster and more accurate, especially for complex functions where graphical analysis might be challenging. However, the graphical method can be a powerful tool for conceptual understanding and for verifying the results obtained algebraically. In practice, a combination of both methods can be beneficial. The graphical method can provide a quick visual check, while the algebraic method ensures accuracy and precision. Ultimately, the choice between the two methods depends on the specific problem, the individual's learning style, and the desired level of understanding.

Conclusion: The Vertex of g(x) = |x-8| + 6 is (8, 6)

In this comprehensive exploration, we have successfully determined the vertex of the absolute value function g(x) = |x-8| + 6 using both graphical and algebraic methods. Through the graphical approach, we visualized the transformations of the basic absolute value function |x|, tracing the shifts caused by the x - 8 and + 6 terms. This visual analysis led us to the conclusion that the vertex is located at (8, 6). The algebraic method provided a more precise solution by recognizing that the minimum value of the absolute value expression |x-8| occurs when x = 8. Substituting this value into the function, we found that g(8) = 6, confirming the vertex at (8, 6). By comparing these two methods, we gained a deeper understanding of the function's behavior and the significance of the vertex. The vertex, in this case, represents the minimum point of the V-shaped graph, indicating the function's lowest value. The coordinates (8, 6) provide valuable information about the function's position and orientation in the coordinate plane. This exploration highlights the importance of understanding both graphical and algebraic techniques in analyzing functions. By mastering these methods, we can effectively solve a wide range of mathematical problems and gain a deeper appreciation for the beauty and power of mathematics. The journey of finding the vertex of g(x) = |x-8| + 6 serves as a testament to the elegance and precision that mathematics offers.

Final Answer: The Vertex of the Graph

Therefore, the final answer is B. (8, 6).