Absolute Value Function Analysis Determining Properties Of F(x) = -(2/3)|x+4| - 6

In this comprehensive analysis, we delve into the intricacies of the absolute value function f(x) = -(2/3)|x+4| - 6, meticulously examining its properties and characteristics to determine the veracity of various statements. Our exploration will encompass the function's vertex, its relationship to the parent function, and the range of its values. By dissecting each aspect, we aim to provide a clear and concise understanding of this function's behavior and graphical representation.

Understanding the Absolute Value Function

The absolute value function, denoted by |x|, is a fundamental concept in mathematics. It returns the non-negative magnitude of a real number, irrespective of its sign. In simpler terms, the absolute value of a number is its distance from zero on the number line. The parent function of absolute value functions is f(x) = |x|, which has a V-shaped graph with its vertex at the origin (0, 0). Transformations applied to this parent function, such as stretches, compressions, reflections, and translations, can alter its shape and position, resulting in a diverse family of absolute value functions.

Analyzing the Given Function: f(x) = -(2/3)|x+4| - 6

The function under scrutiny, f(x) = -(2/3)|x+4| - 6, is a transformed version of the parent absolute value function. To fully comprehend its properties, we must dissect the transformations applied. The coefficient -(2/3) plays a crucial role, as the negative sign indicates a reflection across the x-axis, while the fractional value (2/3) signifies a vertical compression. The term (x+4) within the absolute value function represents a horizontal translation of 4 units to the left, and the constant term -6 denotes a vertical translation of 6 units downward. These transformations collectively mold the graph of the function, shifting its vertex, altering its shape, and influencing its range.

Determining the Vertex

The vertex of an absolute value function is the point where the graph changes direction, forming the characteristic V-shape. For the parent function f(x) = |x|, the vertex resides at the origin (0, 0). However, transformations can displace the vertex. In the given function, f(x) = -(2/3)|x+4| - 6, the horizontal translation of 4 units to the left shifts the vertex's x-coordinate to -4, and the vertical translation of 6 units downward shifts the vertex's y-coordinate to -6. Therefore, the vertex of the function is located at (-4, -6). This understanding of vertex determination is crucial for accurately sketching the graph and comprehending the function's behavior. Recognizing the impact of horizontal and vertical translations on the vertex is a cornerstone of absolute value function analysis. The vertex serves as a key reference point, allowing us to visualize the function's position and orientation on the coordinate plane. Furthermore, the vertex plays a pivotal role in determining the function's range, as it represents the function's minimum or maximum value, depending on whether the graph opens upwards or downwards. In this case, the negative coefficient indicates a downward opening graph, making the vertex the maximum point. Therefore, statement A, which posits the vertex at (-4, 6), is incorrect. A thorough grasp of vertex transformations is essential for anyone seeking to master the intricacies of absolute value functions.

Examining Horizontal Stretch

A horizontal stretch or compression affects the width of the absolute value function's graph. A horizontal stretch expands the graph horizontally, while a horizontal compression squeezes it. These transformations are governed by the coefficient of x within the absolute value function. However, in the given function, f(x) = -(2/3)|x+4| - 6, the coefficient of x inside the absolute value is 1. The 2/3 outside the absolute value represents a vertical compression, not a horizontal stretch. To better understand this, consider the impact of different transformations. A horizontal stretch would involve a coefficient multiplying the x inside the absolute value. For instance, f(x) = |(1/2)x| would represent a horizontal stretch by a factor of 2. In contrast, a vertical compression, as seen in our function, affects the y-values, making the graph appear narrower but not actually stretching it horizontally. Therefore, statement B, which claims a horizontal stretch, is incorrect. This distinction between horizontal and vertical transformations is fundamental to accurately interpreting the function's graphical representation. Visualizing how these transformations alter the shape of the graph is crucial for students to develop a strong intuitive understanding of function transformations. It's important to emphasize that transformations outside the absolute value bars affect vertical aspects, while those inside influence horizontal characteristics. Understanding this core principle allows for accurate identification of stretches, compressions, and reflections.

Determining the Range

The range of a function encompasses all possible output values (y-values). For absolute value functions, the range is influenced by the vertex and the direction in which the graph opens. In our function, f(x) = -(2/3)|x+4| - 6, the negative coefficient indicates that the graph opens downwards. This means the vertex represents the maximum point of the function. Since the vertex is at (-4, -6), the maximum y-value is -6. The graph extends downwards from this point, encompassing all y-values less than or equal to -6. Therefore, the range of the function is y ≤ -6. This is a direct consequence of the downward opening nature of the graph and the y-coordinate of the vertex. To illustrate this further, consider any x-value input into the function. The absolute value part will always be non-negative. Multiplying by -(2/3) will make it non-positive, and subtracting 6 will further reduce the value. Hence, the output will never exceed -6. Understanding the relationship between the leading coefficient, the vertex, and the range is vital for analyzing absolute value functions. The range provides crucial information about the function's behavior, indicating the limits of its output values. Being able to accurately determine the range demonstrates a deep understanding of the function's overall characteristics.

Conclusion

In conclusion, through a meticulous examination of the function f(x) = -(2/3)|x+4| - 6, we have established that the vertex is located at (-4, -6), the graph undergoes a vertical compression rather than a horizontal stretch, and the range is y ≤ -6. Therefore, only the statement concerning the range is correct. This detailed analysis underscores the importance of understanding the individual transformations applied to the parent absolute value function and their cumulative effect on the graph's properties. Mastering these concepts is essential for successfully navigating the world of absolute value functions and their applications.

Which of the following statements accurately describes the function f(x) = -(2/3)|x+4| - 6?

Absolute Value Function Analysis Determining Properties of f(x) = -(2/3)|x+4| - 6