identify the equation for this graph. y = |x - 2| - 3; y = |x - 2| + 3; y = |x + 2| + 3; y = |x + 2| - 3

Explanation:

Step1: Recall the vertex form of absolute - value function

The vertex form of an absolute - value function is (y = a|x - h|+k), where ((h,k)) is the vertex of the V - shaped graph.

Step2: Identify the vertex of the graph

From the graph, the vertex of the absolute - value function is ((- 2,-3)).

Step3: Substitute (h) and (k) into the vertex form

Substituting (h=-2) and (k = - 3) into (y=a|x - h|+k) (assuming (a = 1) since the slope of the right - hand side of the V - shape is (1) and the left - hand side is (-1) for the basic (y = |x|) - like graph). We get (y=|x-(-2)|+(-3)), which simplifies to (y=|x + 2|-3).

Answer:

(y = |x + 2|-3) (the fourth option)