Vertex Of Absolute Value Function G(x) = |x-8|+6 Explained

Determining the vertex of a graph is a fundamental concept in mathematics, particularly when dealing with functions. In this article, we will delve into the specifics of finding the vertex of the graph of the function g(x) = |x - 8| + 6. This problem often appears in algebra and pre-calculus courses, and understanding how to solve it is crucial for grasping the behavior of absolute value functions. We will break down the components of the function, discuss the properties of absolute value graphs, and then pinpoint the vertex. By the end of this guide, you will not only be able to identify the vertex for this specific function but also understand the general principles for finding vertices of similar functions.

The key to identifying the vertex of the graph g(x) = |x - 8| + 6 lies in understanding the nature of absolute value functions and how transformations affect their graphs. The absolute value function, fundamentally, is f(x) = |x|, which produces a V-shaped graph with its vertex at the origin (0, 0). The absolute value ensures that the output is always non-negative, thus creating the characteristic V-shape. When we introduce transformations within the absolute value function, such as in our example g(x) = |x - 8| + 6, the graph undergoes shifts both horizontally and vertically. The term inside the absolute value, (x - 8), dictates a horizontal shift. Specifically, the function |x - 8| shifts the basic absolute value graph 8 units to the right. This is because the expression (x - 8) becomes zero when x = 8, indicating the new horizontal position of the vertex. The + 6 outside the absolute value represents a vertical shift. It moves the entire graph upwards by 6 units. Therefore, the combination of these transformations allows us to pinpoint the vertex of the transformed graph. By recognizing that the horizontal shift is 8 units to the right and the vertical shift is 6 units up, we can confidently determine the vertex. Understanding these transformations is crucial not only for solving this particular problem but also for analyzing a wide range of functions in mathematics.

The given function g(x) = |x - 8| + 6 represents a transformation of the basic absolute value function. To accurately determine the vertex, we must understand how horizontal and vertical shifts influence the graph. The general form of an absolute value function with transformations is g(x) = a|x - h| + k, where (h, k) represents the vertex of the graph. In this form, h dictates the horizontal shift, and k dictates the vertical shift. When we compare our given function, g(x) = |x - 8| + 6, to the general form, we can identify the values of h and k. Here, h = 8 and k = 6. This means the graph of the basic absolute value function |x| has been shifted 8 units to the right and 6 units upwards. Therefore, the vertex of the graph of g(x) = |x - 8| + 6 is the point (8, 6). This understanding is vital for visualizing and analyzing the behavior of absolute value functions. Knowing the vertex allows us to quickly sketch the graph and understand its key features, such as its symmetry and range. Moreover, this approach is applicable to any absolute value function, making it a valuable tool in mathematical problem-solving. By mastering the concept of transformations and their effect on the vertex, students can confidently tackle more complex problems involving absolute value functions. The ability to identify the vertex also lays the groundwork for understanding other transformations, such as stretches and reflections, which can further modify the shape and position of the graph. In conclusion, the vertex serves as a critical reference point for analyzing absolute value functions, and understanding how to find it is essential for success in algebra and beyond.

Detailed Explanation of the Solution

To find the vertex of the graph of the function g(x) = |x - 8| + 6, we need to understand the transformations applied to the basic absolute value function, f(x) = |x|. The vertex of the basic absolute value function is at the origin, (0, 0). Transformations to the function will shift this vertex. In the given function, we have two transformations: a horizontal shift and a vertical shift. The expression (x - 8) inside the absolute value indicates a horizontal shift. Specifically, it shifts the graph 8 units to the right. This is because the function reaches its minimum value when x - 8 = 0, which occurs when x = 8. The + 6 outside the absolute value indicates a vertical shift. It shifts the entire graph 6 units upwards. Therefore, the vertex of the transformed graph will be the result of these shifts applied to the original vertex (0, 0). Shifting (0, 0) eight units to the right gives us (8, 0), and then shifting six units upwards gives us (8, 6). Thus, the vertex of the graph of g(x) = |x - 8| + 6 is (8, 6). This understanding is crucial for quickly identifying the vertices of various absolute value functions. By recognizing the horizontal and vertical shifts, we can easily determine the vertex without having to graph the function. This approach saves time and provides a clear understanding of the function's behavior. In more complex scenarios, where additional transformations such as stretches or reflections are involved, the same principles apply. Identifying each transformation and its effect on the vertex is key to accurately determining the final vertex position. Ultimately, mastering these concepts enables students to confidently analyze and manipulate absolute value functions, laying a strong foundation for advanced mathematical topics.

Analyzing the transformations applied to the absolute value function is paramount in determining the vertex of the graph. The given function, g(x) = |x - 8| + 6, builds upon the foundational absolute value function, f(x) = |x|, which has its vertex at (0, 0). The transformation (x - 8) within the absolute value bars signifies a horizontal translation. A critical point to remember is that (x - h) translates the graph h units to the right if h is positive and |h| units to the left if h is negative. In our case, (x - 8) indicates a shift of 8 units to the right because h = 8. This shift occurs because the minimum value of the absolute value expression is achieved when x - 8 = 0, solving for x yields x = 8. This x-value represents the horizontal coordinate of the new vertex. The second transformation is the addition of 6 outside the absolute value, which represents a vertical translation. Adding a constant k to the function shifts the graph k units upward if k is positive and |k| units downward if k is negative. In g(x) = |x - 8| + 6, the + 6 shifts the graph 6 units upward. This shift directly affects the y-coordinate of the vertex. Combining both transformations, the original vertex (0, 0) is shifted 8 units to the right and 6 units upward, resulting in the new vertex (8, 6). This methodical approach to analyzing transformations is not only effective for absolute value functions but also applies to other types of functions, such as quadratics, exponentials, and trigonometric functions. The ability to identify and interpret these transformations is a cornerstone of understanding function behavior and graphical representation in mathematics.

The horizontal and vertical shifts are the primary determinants of the vertex location in transformed absolute value functions. Let's further illustrate how these shifts affect the original vertex (0, 0) of the basic absolute value function f(x) = |x|. Consider the term (x - 8) inside the absolute value of g(x) = |x - 8| + 6. This term dictates the horizontal shift. To find the x-coordinate of the vertex, we set the expression inside the absolute value equal to zero and solve for x: x - 8 = 0. This gives us x = 8. Therefore, the graph is shifted 8 units to the right. This means that the x-coordinate of the vertex of g(x) will be 8. Next, we consider the term + 6 outside the absolute value. This term dictates the vertical shift. A positive value indicates an upward shift, and a negative value indicates a downward shift. In this case, + 6 signifies a shift of 6 units upwards. This means that the y-coordinate of the vertex of g(x) will be 6. Combining these two shifts, we start with the original vertex (0, 0), shift it 8 units to the right to reach (8, 0), and then shift it 6 units upwards to reach (8, 6). Therefore, the vertex of the graph of g(x) = |x - 8| + 6 is (8, 6). Understanding this process allows for quick identification of the vertex without needing to graph the function or use complex calculations. This method is particularly valuable in test-taking situations where time efficiency is crucial. Moreover, this principle of analyzing horizontal and vertical shifts is a fundamental concept in function transformations, applicable across various types of functions in mathematics.

Conclusion: Identifying the Vertex (8, 6)

In conclusion, to determine the vertex of the graph of g(x) = |x - 8| + 6, we have analyzed the transformations applied to the basic absolute value function, f(x) = |x|. The function g(x) = |x - 8| + 6 involves a horizontal shift of 8 units to the right, dictated by the (x - 8) term inside the absolute value, and a vertical shift of 6 units upwards, dictated by the + 6 term outside the absolute value. Starting from the vertex of the basic absolute value function at (0, 0), these transformations shift the vertex to (8, 6). Therefore, the vertex of the graph of g(x) = |x - 8| + 6 is (8, 6). This understanding is crucial for sketching the graph of the function and analyzing its behavior. By recognizing the effects of horizontal and vertical shifts, one can easily identify the vertex and understand the overall shape and position of the graph. This approach is a fundamental concept in algebra and pre-calculus, applicable to a wide range of functions beyond just absolute value functions. Mastering this skill allows for a deeper comprehension of function transformations and their impact on graphical representations. Furthermore, the ability to quickly identify the vertex is a valuable tool in problem-solving, providing a key piece of information for analyzing and manipulating functions effectively. The process we have outlined here provides a systematic method for determining the vertex of any transformed absolute value function, making it an essential technique for students and anyone working with mathematical functions.