Question

which of the following is the graph of this absolute value function? $y = |x - 1|$

Explanation:

Step1: Recall the vertex form of absolute value functions

The general form of an absolute value function is \( y = |x - h| + k \), where \((h, k)\) is the vertex of the V - shaped graph. For the function \( y=|x - 1|\), we can rewrite it as \( y=|x - 1|+0 \), so the vertex of the graph should be at the point \((h,k)=(1,0)\).

Step2: Analyze the x - intercept (vertex) of each graph

• For the first graph: The vertex is at \((- 1,0)\), which does not match the vertex \((1,0)\) of \(y = |x - 1|\).
• For the second graph: The vertex is at \((0,-1)\), which does not match the vertex \((1,0)\) of \(y = |x - 1|\).
• For the third graph: The vertex is at \((1,0)\), which matches the vertex of the function \(y=|x - 1|\). Also, we can check the slope of the two arms of the V - shape. For \(y = |x - 1|\), when \(x>1\), \(y=x - 1\) (slope \(m = 1\)), and when \(x<1\), \(y=-(x - 1)=-x + 1\) (slope \(m=- 1\)), which is consistent with the shape of the third graph.

Answer:

The third graph (the one with vertex at (1,0) and x - axis labels - 2, - 1, 0, 1, 2, 3, 4 and y - axis labels - 1, 1, 2, 3, 4)