which graph represents the function $p(x) = |x - 1|$?

Explanation:

Step1: Recall the vertex of absolute - value function

The general form of an absolute - value function is (y = |x - h|+k), where the vertex of the graph is at the point ((h,k)). For the function (p(x)=|x - 1|), we can compare it with the general form (y=|x - h|+k). Here, (h = 1) and (k = 0), so the vertex of the graph of (p(x)) should be at the point ((1,0)).

Step2: Analyze the slope of the two parts of the absolute - value function

The absolute - value function (y = |x - 1|) can be written as a piece - wise function:

• When (x-1\geq0) (i.e., (x\geq1)), (p(x)=x - 1). The slope of the line (y=x - 1) is (m = 1) (since the equation is in the form (y=mx + b) with (m = 1) and (b=- 1)).
• When (x - 1<0) (i.e., (x<1)), (p(x)=-(x - 1)=-x + 1). The slope of the line (y=-x + 1) is (m=-1).

Step3: Match the graph with the vertex and slopes

We need to find the graph that has its vertex at ((1,0)), a slope of (1) for (x\geq1) and a slope of (- 1) for (x<1).

Looking at the options, we check the vertex first. The graph with vertex at ((1,0)) (and with a slope of (1) for (x\geq1) and (-1) for (x < 1)) is the one where the "corner" of the V - shaped graph is at ((1,0)).

Answer:

The graph (the fourth one, assuming the fourth graph has its vertex at ((1,0)) and the correct slopes for the two branches of the absolute - value function) represents the function (p(x)=|x - 1|). If we assume the graphs are labeled from top - left, top - middle, top - right, bottom - right, the bottom - right graph (the one with vertex at ((1,0))) is the correct one.