select the graph of $y = \\frac{1}{2}|x| - 2$.
select the graph of $y = \\frac{1}{2}|x| - 2$.
Answer
Explanation:
Step1: Analyze the vertex of the function
The general form of an absolute - value function is (y = a|x - h|+k), where ((h,k)) is the vertex. For the function (y=\frac{1}{2}|x|-2), (h = 0) and (k=-2). So the vertex is ((0, - 2)).
Step2: Analyze the slope for (x\geq0)
When (x\geq0), (y=\frac{1}{2}x - 2). The slope (m=\frac{1}{2}).
Step3: Analyze the slope for (x<0)
When (x < 0), (y=\frac{1}{2}(-x)-2=-\frac{1}{2}x - 2). The slope (m =-\frac{1}{2}).
Answer:
The graph with vertex ((0,-2)) and slopes (\pm\frac{1}{2}) (the second graph in the bottom - left among the four graphs shown in the problem, assuming the standard order of graphs in a (2\times2) grid where we check the vertex and slopes) is the correct graph of (y = \frac{1}{2}|x|-2).