Finding The Vertex Of F(x) = |x-13| + 11 A Step-by-Step Guide

In the realm of mathematics, understanding the properties of functions is crucial for solving a wide array of problems. Among these functions, absolute value functions hold a unique position due to their distinctive V-shaped graphs. The vertex of such a graph represents a critical point, marking either the minimum or maximum value of the function. In this comprehensive guide, we will delve into the intricacies of absolute value functions and explore how to determine the vertex of the graph of f(x) = |x-13| + 11. We will also analyze the given options and arrive at the correct answer, providing a step-by-step explanation to ensure a clear understanding of the underlying concepts. This detailed exploration will not only equip you with the knowledge to solve this specific problem but also provide a solid foundation for tackling similar problems involving absolute value functions. By the end of this guide, you will have a thorough grasp of the characteristics of absolute value functions and the significance of the vertex in their graphical representation. Let's embark on this mathematical journey together, unraveling the secrets of absolute value functions and mastering the art of finding their vertices.

Decoding Absolute Value Functions

To effectively determine the vertex of the graph of f(x) = |x-13| + 11, we must first establish a solid understanding of absolute value functions in general. An absolute value function is a function that contains an algebraic expression within absolute value symbols. The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that the absolute value of a number is always non-negative. For example, |5| = 5 and |-5| = 5. The general form of an absolute value function is f(x) = a|x - h| + k, where a, h, and k are constants. The graph of an absolute value function is always V-shaped, with the vertex representing the point where the graph changes direction. The constant a determines the direction and steepness of the V. If a is positive, the V opens upwards, and if a is negative, the V opens downwards. The constants h and k determine the position of the vertex. The vertex of the graph is located at the point (h, k). Understanding these fundamental properties of absolute value functions is essential for identifying and interpreting their graphs. The absolute value function's unique behavior stems from its piecewise definition, where the function's output is either the input itself (if the input is non-negative) or the negation of the input (if the input is negative). This piecewise nature is what gives rise to the characteristic V-shape of the graph. Mastering these concepts will enable you to confidently analyze and manipulate absolute value functions in various mathematical contexts.

Identifying the Vertex Form of the Given Function

Now that we have a firm grasp of the fundamentals of absolute value functions, let's focus on the specific function presented in the problem: f(x) = |x-13| + 11. To determine the vertex of this function's graph, we need to recognize that it is already presented in the vertex form of an absolute value function, which is f(x) = a|x - h| + k. In this form, the vertex of the graph is readily identifiable as the point (h, k). By carefully comparing the given function f(x) = |x-13| + 11 with the general vertex form, we can extract the values of h and k. Notice that the coefficient a in front of the absolute value term is 1 in this case, indicating that the V-shape will open upwards. The expression inside the absolute value, x - 13, corresponds to the term x - h in the general form. Therefore, we can deduce that h = 13. The constant term added outside the absolute value, +11, corresponds to the constant k in the general form. Thus, we have k = 11. With the values of h and k determined, we can confidently state that the vertex of the graph of f(x) = |x-13| + 11 is located at the point (13, 11). This straightforward process of comparing the given function with the vertex form highlights the importance of understanding the standard forms of mathematical expressions. By recognizing these forms, we can quickly extract key information and solve problems efficiently. In this case, the vertex form directly reveals the coordinates of the vertex, simplifying the task of finding this crucial point on the graph.

Determining the Vertex: A Step-by-Step Approach

To solidify our understanding and ensure we arrive at the correct answer, let's walk through a step-by-step approach to determining the vertex of the graph of f(x) = |x-13| + 11. This methodical approach will reinforce the concepts we've discussed and provide a clear framework for tackling similar problems.

1. Identify the vertex form: As we established earlier, the function f(x) = |x-13| + 11 is already in the vertex form f(x) = a|x - h| + k. This is the crucial first step, as it allows us to directly identify the values of h and k. Recognizing the vertex form simplifies the process significantly.
2. Extract the values of h and k: By comparing the given function with the vertex form, we can readily identify h and k. In this case, h = 13 and k = 11. Remember that the h value is the x-coordinate of the vertex and is the value that makes the expression inside the absolute value equal to zero. The k value is the y-coordinate of the vertex and represents the minimum or maximum value of the function.
3. State the vertex: The vertex of the graph is the point (h, k). With h = 13 and k = 11, the vertex is (13, 11). This is the key result we've been aiming for.
4. Verify the result: To ensure accuracy, it's always a good practice to verify the result. We can do this by considering the nature of the absolute value function. Since the coefficient of the absolute value term is positive (1 in this case), the graph opens upwards. This means that the vertex represents the minimum point of the function. At x = 13, the absolute value term becomes zero, and the function attains its minimum value of 11. This confirms that (13, 11) is indeed the vertex.

By following these steps, we can confidently determine the vertex of any absolute value function expressed in vertex form. This systematic approach not only helps in solving problems accurately but also fosters a deeper understanding of the underlying concepts.

Analyzing the Answer Choices

Now that we have determined the vertex of the graph of f(x) = |x-13| + 11 to be (13, 11), let's analyze the given answer choices to identify the correct option. This step is crucial to ensure we select the appropriate answer from the provided alternatives.

The answer choices are:

A. (-11, 13) B. (-13, 11) C. (11, 13) D. (13, 11)

By comparing our calculated vertex (13, 11) with the answer choices, it becomes immediately clear that option D matches our result. The other options present incorrect coordinates for the vertex. Option A has the coordinates reversed and with incorrect signs. Option B has an incorrect x-coordinate, and option C has incorrect coordinates as well. Therefore, we can confidently eliminate options A, B, and C. This process of elimination is a valuable strategy in multiple-choice questions, allowing us to narrow down the possibilities and increase our chances of selecting the correct answer. In this case, by systematically determining the vertex and then comparing it with the options, we have arrived at the correct answer with certainty. This reinforces the importance of both understanding the underlying concepts and employing effective problem-solving techniques.

Conclusion: The Vertex and its Significance

In conclusion, the vertex of the graph of f(x) = |x-13| + 11 is (13, 11). This point represents the minimum value of the function and is a crucial characteristic of the graph. Throughout this guide, we have explored the concept of absolute value functions, identified the vertex form, and developed a step-by-step approach to determine the vertex. We also analyzed the answer choices and confirmed our result. Understanding the vertex of an absolute value function is essential for graphing and analyzing these functions. The vertex provides key information about the function's behavior, such as its minimum or maximum value and its axis of symmetry. The x-coordinate of the vertex represents the axis of symmetry, which is a vertical line that divides the V-shaped graph into two symmetrical halves. The y-coordinate of the vertex represents the minimum or maximum value of the function, depending on whether the graph opens upwards or downwards. In the case of f(x) = |x-13| + 11, the vertex (13, 11) indicates that the minimum value of the function is 11, which occurs at x = 13. This comprehensive understanding of absolute value functions and their vertices is a valuable asset in mathematics and related fields. By mastering these concepts, you can confidently tackle a wide range of problems involving absolute value functions and their graphical representations. The ability to determine the vertex quickly and accurately is a fundamental skill that will serve you well in your mathematical journey.