Problem

Infer the \(\text{single}\) motion from this allowed \(\text{one}\text{-}\text{step}\) family: \(\text{one}\) translation, \(\text{one}\) reflection across a coordinate axis, or \(\text{one}\) \(90°/180°/270°\) rotation about the origin. It maps \(A(1,0)\), \(B(2,0)\), \(C(1,3)\) to \(A^{\prime}(0,1)\), \(B^{\prime}(0,2)\), \(C^{\prime}(-3,1)\). Give the motion, coordinate rule, and checks of \(\text{all}~\text{three}~\text{mappings}\).

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motion = reflection across y-axis; rule = (x,y)→(-x,y); checks = A→(-1,0), B→(-2,0), C→(-1,3)

motion = 90° clockwise rotation; rule = (x,y)→(y,-x); checks = A→(0,-1), B→(0,-2), C→(3,-1)

motion = 90° counterclockwise rotation about the origin; rule = (x,y)→(-y,x); checks = A→A′, B→B′, C→C′

motion = translation by ⟨-1,1⟩; rule = (x,y)→(x-1,y+1); checks = A→(0,1), B→(1,1), C→(0,4)

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