which of the following is the graph of $f(x)=-0.5|x + 3|-2$?
which of the following is the graph of $f(x)=-0.5|x + 3|-2$?
Answer
Explanation:
Step1: Analyze the vertex form of absolute - value function
The general form of an absolute - value function is (y = a|x - h|+k), and its vertex is ((h,k)). For the function (f(x)=- 0.5|x + 3|-2), we have (a=-0.5), (h=-3), and (k = - 2). So the vertex of the function is ((-3,-2)).
Step2: Analyze the slope of the rays
Since (a=-0.5\lt0), the graph opens downwards. When (x\gt - 3), (f(x)=-0.5(x + 3)-2=-0.5x-1.5 - 2=-0.5x-3.5), and the slope of the right - hand ray is (-0.5). When (x\lt - 3), (f(x)=-0.5(-(x + 3))-2 = 0.5x+1.5-2=0.5x - 0.5), and the slope of the left - hand ray is (0.5).
Answer:
We need to find the graph with vertex at ((-3,-2)) and opening downwards. Without seeing all the options clearly, we know that the correct graph should have a vertex at the point ((-3,-2)) and open downwards due to the negative coefficient of the absolute - value part. If there is a graph among the options with vertex ((-3,-2)) and opening downwards, that is the correct one.