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 \(A(1,2)\), \(B(4,2)\), \(C(1,5)\) to \(A^{\prime}(-1,2)\), \(B^{\prime}(-4,2)\), \(C^{\prime}(-1,5)\). Give the motion, coordinate rule, and checks of \(\text{all}~\text{three}~\text{mappings}\).

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motion = reflection across x-axis; rule = (x,y)→(x,-y); checks = A→(1,-2), B→(4,-2), C→(1,-5)

motion = 90° counterclockwise rotation; rule = (x,y)→(-y,x); checks = A→(-2,1), B→(-2,4), C→(-5,1)

motion = translation by ⟨-2,0⟩; rule = (x,y)→(x-2,y); checks = A→(-1,2), B→(2,2), C→(-1,5)

motion = reflection across the y-axis; rule = (x,y)→(-x,y); checks = A→A′, B→B′, C→C′

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