Identifying the Correct Graph for r(x) = |x – 2| – 1

Identifying the correct graph for the function ( r(x) = |x – 2| – 1 ) requires a nuanced understanding of the components that define the absolute value function, its transformations, and the implications these have on the overall shape of the graph. This function, which involves the absolute value of a linear expression, has unique characteristics that can significantly influence its graphical representation. In this article, we will dissect the components of this function and provide a critical analysis of how to choose the appropriate graph that accurately represents it.

Understanding the Components of r(x) = |x – 2| – 1

To begin with, the function ( r(x) = |x – 2| – 1 ) can be broken down into two distinct parts: the absolute value expression ( |x – 2| ) and the vertical shift represented by the subtraction of 1. The absolute value function itself, ( |x – 2| ), reflects the distance of any number ( x ) from the point 2 on the x-axis. This creates a V-shaped graph that opens upwards, with its vertex located at the point (2, 0). Therefore, the basic shape of the graph is influenced heavily by the location of this vertex.

Next, the subtraction of 1 from the absolute value expression indicates a vertical shift downward by one unit. Consequently, while the vertex of the basic absolute value function is at (2, 0), the vertex of ( r(x) ) will be shifted to (2, -1). This transformation is critical, as it alters the position of the graph on the Cartesian plane, thereby affecting how we perceive the function’s behavior. Moreover, the slopes of the lines forming the V will remain unchanged at 1 and -1, but their starting point will now be at a lower level.

In summary, understanding the components of ( r(x) ) involves recognizing the effect of the absolute value and its transformations. The absolute value function creates a V-shape, and the vertical shift modifies its position on the graph. Therefore, a deep comprehension of these components is essential for accurately identifying the correct graph representation of ( r(x) = |x – 2| – 1 ).

Choosing the Right Graph: A Critical Analysis of r(x)

When tasked with choosing the correct graph for ( r(x) = |x – 2| – 1 ), it is essential to look beyond mere visual inspection. A critical analysis should involve consideration of key features such as intercepts, vertex position, and overall shape. The vertex, as established earlier, is at (2, -1), which is a crucial point of reference for graphing. The graph will also cross the x-axis at points that satisfy the equation ( |x – 2| – 1 = 0 ), which can be solved to find the x-intercepts located at ( x = 1 ) and ( x = 3 ).

Additionally, it is vital to evaluate the slope of the lines extending from the vertex. The left side of the V will decrease with a slope of -1 for ( x < 2 ), and the right side will increase with a slope of 1 for ( x > 2 ). This means that any graph that incorrectly represents these slopes or misplaces the vertex will ultimately fail to accurately depict the function. Thus, the key to identifying the right graph lies in ensuring that all these characteristics align with the expected features of the absolute value function and its transformations.

Furthermore, the overall behavior of the function as it approaches both ends of the x-axis should be considered. As ( x ) moves away from the vertex, the function will trend upwards indefinitely in both directions, which is a defining characteristic of V-shaped graphs derived from absolute value functions. Consequently, any chosen graph should exhibit this behavior while maintaining the critical points and slopes discussed. Only by conducting such a thorough analysis can one arrive at the correct graph for ( r(x) = |x – 2| – 1 ).

In conclusion, identifying the correct graph for the function ( r(x) = |x – 2| – 1 ) is an exercise in understanding the intricacies of absolute value functions and their transformations. By dissecting the components of the function and analyzing its graphical representation through a critical lens, one can arrive at a precise understanding of its key features, including the vertex, slopes, intercepts, and overall shape. Ultimately, this comprehensive approach is essential for accurately selecting and interpreting the graph of ( r(x) ), paving the way for greater insights into the behavior of absolute value functions in mathematics.