Short Answer 
The process of transforming the absolute value function begins with identifying its basic “V” shape with the equation y = |x|. In the function f(x) = -0.5|x+3| -2, the transformations include shifting left 3 units, reflecting over the x-axis, compressing vertically by a factor of 2, and moving down 2 units, resulting in a vertex positioned at (-3, -2) for an inverted “V” shape.
Step 1: Identify the Basic Shape
Start with the basic absolute value function, represented as y = |x|. This function produces a distinct “V” shape in its graph. Understanding this basic shape is crucial as it will serve as the foundation for all subsequent transformations applied to the function.
Step 2: Apply Transformations
The function f(x) = -0.5|x+3| -2 involves several transformations. First, shift the graph 3 units to the left by adjusting the input inside the absolute value. Next, reflect it over the x-axis (which inverts the “V” shape) due to the negative coefficient, and compress it vertically by a factor of 2. Finally, subtract 2, shifting the entire graph downwards.
Step 3: Determine the Vertex and Final Position
The vertex of the transformed graph can be found at the point (-3, -2), which is the highest point of the inverted “V.” After applying all transformations, the graph will open downwards and appear as an upside-down “V” shape. Remember that this vertex marks the tip of the graph, indicating where it changes direction.