Which graph represents the function $r(x)=|x-2|-1$
Answer (1)
The function is r ( x ) = ∣ x − 2∣ − 1 .
The graph of ∣ x − 2∣ is the graph of ∣ x ∣ shifted 2 units to the right.
The graph of ∣ x − 2∣ − 1 is the graph of ∣ x − 2∣ shifted 1 unit downwards.
The vertex of the graph is at ( 2 , − 1 ) .
The graph is a V-shape with the vertex at ( 2 , − 1 ) .
Explanation
Analyze the function We are given the function r ( x ) = ∣ x − 2∣ − 1 and we want to determine its graph. This is an absolute value function, which has a V-shape. The basic absolute value function y = ∣ x ∣ has its vertex at the origin ( 0 , 0 ) . The given function is a transformation of the basic absolute value function.
Horizontal shift The transformation xv er t x − 2 shifts the graph horizontally. Specifically, the graph of ∣ x − 2∣ is the graph of ∣ x ∣ shifted 2 units to the right. This means the vertex of ∣ x − 2∣ is at x = 2 , so the vertex is at ( 2 , 0 ) .
Vertical shift The transformation ∣ x − 2∣ v er t ∣ x − 2∣ − 1 shifts the graph vertically. Specifically, the graph of ∣ x − 2∣ − 1 is the graph of ∣ x − 2∣ shifted 1 unit downwards. This means the vertex of ∣ x − 2∣ − 1 is at ( 2 , − 1 ) .
Consider two cases To further understand the graph, we can consider two cases:
Case 1: x ≥ 2 . In this case, ∣ x − 2∣ = x − 2 , so r ( x ) = x − 2 − 1 = x − 3 .
Case 2: x < 2 . In this case, ∣ x − 2∣ = − ( x − 2 ) = − x + 2 , so r ( x ) = − x + 2 − 1 = − x + 1 .
Sketch the graph So, for x ≥ 2 , r ( x ) = x − 3 , which is a line with slope 1 and y-intercept -3. For x < 2 , r ( x ) = − x + 1 , which is a line with slope -1 and y-intercept 1. The vertex of the graph is at ( 2 , − 1 ) . The graph is a V-shape with the vertex at ( 2 , − 1 ) .
Final Answer The graph of r ( x ) = ∣ x − 2∣ − 1 is a V-shaped graph with vertex at ( 2 , − 1 ) . The graph has a slope of -1 for x < 2 and a slope of 1 for x ≥ 2 .
Examples
Absolute value functions are used in many real-world applications, such as calculating distances or errors. For example, if you want to find the difference between a measured value and an expected value, you can use the absolute value function to ensure that the difference is always positive. In engineering, absolute value functions are used to model tolerances and deviations from desired specifications. Understanding transformations of absolute value functions helps in analyzing and predicting the behavior of systems where deviations from a set point are important.
