Graphing Absolute Value Functions (with Transformations)

Draw a graph of an absolute value function with transformations in a coordinate plane, using geometric aids to clarify how the changes affect the shape. First choose key points that define the characteristic vertex and the two end locations of the V-shaped, piecewise structure, then connect the appropriate endpoints with segments to form the reference lines. From the central point, construct vectors toward the endpoints to indicate the transformation directions, and use a polygonal region built from the key points to visually group the upper and lower parts of the graph. Add rotated text near the center to label the parameter orientation. Include on-plot check messages indicating whether the horizontal shift, vertical stretch, and vertical shift match the chosen conditions.