Analyzing F(x) + 2 = (1/6)|x - 3|: Which Statement Is True?

by Alex Johnson 60 views

Let's dive into the world of functions and explore the characteristics of the function f(x) + 2 = (1/6)|x - 3|. In this article, we'll break down this function, analyze its components, and determine which statement accurately describes it. We'll cover key aspects like the vertex, transformations, and direction of opening, providing a comprehensive understanding of the function's behavior. So, grab your mathematical thinking caps, and let's get started!

Understanding the Function

To accurately describe the function f(x) + 2 = (1/6)|x - 3|, we first need to isolate f(x). Subtracting 2 from both sides of the equation, we get:

f(x) = (1/6)|x - 3| - 2

Now, let's break down this function piece by piece. This form of equation reveals the transformations applied to the absolute value parent function, |x|. The absolute value function is a cornerstone of many mathematical concepts, and understanding its transformations is crucial for analyzing more complex functions. In this instance, the parent function, |x|, undergoes several key transformations that ultimately define the behavior and graphical representation of f(x). These transformations include horizontal shifts, vertical stretches or compressions, and vertical translations. Each of these alterations affects the parent function in a unique way, resulting in a modified graph with distinct characteristics. Identifying and understanding these transformations is essential for accurately interpreting and analyzing functions in mathematics. Recognizing the role of each transformation allows us to predict the function's behavior and visualize its graphical representation with precision. This deeper comprehension of functional transformations not only enriches our understanding of mathematical principles but also enhances our problem-solving capabilities in diverse mathematical contexts. So, let's delve further into the specifics of these transformations to gain a complete understanding of how they contribute to the overall behavior of the function f(x). By carefully analyzing each transformation, we can piece together the puzzle and gain a comprehensive perspective on the function's characteristics.

Identifying Key Transformations

  • (1/6): This represents a vertical compression by a factor of 1/6. Imagine taking the graph of |x| and squeezing it vertically, making it wider.
  • (x - 3): This indicates a horizontal shift of 3 units to the right. The graph is moved along the x-axis, changing its position but maintaining its fundamental shape. This shift is a crucial component of understanding the function's behavior, particularly in relation to its vertex and symmetry. The horizontal shift directly impacts the x-coordinate of the vertex, which serves as a critical reference point for the function. Understanding this horizontal movement is essential for accurately interpreting and analyzing the function's graphical representation. By recognizing the magnitude and direction of the horizontal shift, we can readily determine the vertex's position and gain insights into the function's overall behavior. This knowledge is not only valuable for graphing the function but also for solving related mathematical problems and applications.
  • -2: This represents a vertical shift of 2 units downward. The entire graph is moved down along the y-axis. This shift affects the vertical position of the graph, particularly the y-coordinate of the vertex and the overall range of the function. The vertical shift is a significant transformation that alters the function's placement in the coordinate plane. By understanding the magnitude and direction of this shift, we can accurately determine the vertical position of the vertex and the function's minimum or maximum value. This information is crucial for analyzing the function's behavior and solving problems related to its vertical displacement. The vertical shift, along with other transformations, collectively shapes the function's graph and contributes to its unique characteristics.

Analyzing the Statements

Now that we've broken down the function, let's analyze the given statements:

A. The graph of f(x) has a vertex of (-3, 2).

The vertex of the parent function |x| is at (0, 0). With the transformations we identified:

  • Horizontal shift 3 units to the right changes the x-coordinate to +3.
  • Vertical shift 2 units downward changes the y-coordinate to -2.

Therefore, the vertex of f(x) is at (3, -2). This eliminates option A.

The vertex of a function is a crucial point that provides insights into the function's behavior and graph. It represents the function's minimum or maximum value, depending on the direction in which the graph opens. In the case of absolute value functions, the vertex is the point where the graph changes direction, forming a characteristic V-shape. Understanding the vertex is essential for analyzing the function's properties, including its domain, range, and symmetry. The vertex also plays a significant role in determining the function's transformations, such as horizontal and vertical shifts. By identifying the vertex, we can gain a deeper understanding of the function's graphical representation and its relationship to the parent function. Moreover, the vertex is a key element in solving optimization problems, where we seek to find the maximum or minimum value of a function. Its importance in various mathematical contexts underscores the significance of understanding and analyzing the vertex of a function. Whether it's used for graphing, problem-solving, or theoretical analysis, the vertex serves as a fundamental tool in the study of functions.

B. The graph of f(x) is a horizontal compression of the graph of the parent function.

The factor of 1/6 is a vertical compression, not a horizontal one. A horizontal compression would involve a coefficient inside the absolute value, affecting the x-variable directly. Thus, statement B is incorrect. Vertical compression and horizontal compression are distinct transformations that affect the graph of a function in different ways. Vertical compression involves scaling the function's y-values, while horizontal compression involves scaling the x-values. Understanding the difference between these transformations is crucial for accurately interpreting and analyzing function graphs. Vertical compression squeezes the graph vertically, making it appear wider, while horizontal compression squeezes the graph horizontally, making it appear taller. The coefficients that determine the amount of compression or stretching are applied differently for each type of transformation. For vertical compression, the coefficient is multiplied by the function's output, whereas for horizontal compression, the coefficient is multiplied by the function's input. Recognizing these distinctions allows us to effectively analyze and manipulate function graphs to solve mathematical problems and gain insights into the behavior of different functions.

C. The graph of f(x) opens downward.

The coefficient 1/6 in front of the absolute value is positive. A negative coefficient would be required for the graph to open downward. Therefore, statement C is incorrect. The direction in which a graph opens is a fundamental characteristic that provides valuable information about the function's behavior. For absolute value functions, the graph opens upward if the coefficient in front of the absolute value term is positive, and it opens downward if the coefficient is negative. This characteristic is directly related to the concavity of the graph, with upward-opening graphs being concave up and downward-opening graphs being concave down. Understanding the direction of opening is essential for determining whether the function has a minimum or maximum value and for sketching the graph accurately. It also plays a crucial role in solving optimization problems and analyzing the function's end behavior. By simply observing the sign of the leading coefficient, we can quickly determine the direction in which the graph opens, which is a key step in understanding the function's overall behavior and properties.

D. The graph of f(x) is a vertical compression and a translation of the graph of the parent function.

This statement accurately describes the transformations we identified: a vertical compression by a factor of 1/6 and translations (horizontal and vertical shifts). Therefore, statement D is the correct answer.

Conclusion

After analyzing the function f(x) + 2 = (1/6)|x - 3|, we determined that the correct statement is D. The graph of f(x) is a vertical compression and a translation of the graph of the parent function. We arrived at this conclusion by carefully examining the transformations applied to the parent function |x|, including the vertical compression, horizontal shift, and vertical shift. Understanding these transformations allowed us to accurately describe the characteristics of the function and identify the correct statement. By breaking down the function into its components and analyzing each transformation, we gained a comprehensive understanding of its behavior and graphical representation. This approach is essential for solving mathematical problems and gaining deeper insights into the world of functions.

For further exploration of absolute value functions and their properties, you can visit resources like Khan Academy.