1. The problem asks to identify the algebraic transformation that maps the first square with vertices \((-4,7), (-1,7), (-1,4), (-4,4)\) to the second square with vertices \((2,3), (5,3), (5,6), (2,6)\). 2. Let's analyze the transformation options given: - \((x, y) \to (y, -x)\) - \((x, y) \to (-y, x)\) - \((x, y) \to (-x, y)\) - \((x, y) \to (-x, -y)\) 3. To find the correct transformation, apply each to one vertex of the first square and check if it matches a vertex of the second square. 4. Take vertex \(E = (-4,7)\) from the first square. - For \((x, y) \to (y, -x)\): $$ (y, -x) = (7, 4) $$ This is not a vertex of the second square. - For \((x, y) \to (-y, x)\): $$ (-y, x) = (-7, -4) $$ Not a vertex of the second square. - For \((x, y) \to (-x, y)\): $$ (-x, y) = (4, 7) $$ Not a vertex of the second square. - For \((x, y) \to (-x, -y)\): $$ (-x, -y) = (4, -7) $$ Not a vertex of the second square. 5. None of these match directly, so consider that the square might have been translated after rotation or reflection. 6. Check the transformation \((x, y) \to (y, -x)\) plus a translation vector: - Applying \((x, y) \to (y, -x)\) to \((-4,7)\) gives \((7,4)\). - The second square's vertex \(E\) is at \((2,3)\). - The difference is \((2-7, 3-4) = (-5, -1)\). 7. Apply the same transformation plus translation \((-5, -1)\) to all vertices: - \((-1,7)\) maps to \((7,1)\) plus translation \((-5,-1)\) = \((2,0)\) which does not match second square. 8. Try transformation \((x, y) \to (-y, x)\) plus translation: - \((-4,7)\) maps to \((-7,-4)\). - Second square vertex \(E\) is \((2,3)\). - Translation vector needed: \((2+7, 3+4) = (9,7)\). - Apply to \((-1,7)\): \((-7,-1) + (9,7) = (2,6)\) matches vertex \(D\) of second square. - Apply to \((-1,4)\): \((-4,-1) + (9,7) = (5,6)\) matches vertex \(G\). - Apply to \((-4,4)\): \((-4,-4) + (9,7) = (5,3)\) matches vertex \(F\). 9. This confirms the transformation is \((x, y) \to (-y, x)\) plus translation \((9,7)\). 10. Since the question asks for the algebraic representation of the transformation (ignoring translation), the correct algebraic transformation is: $$ (x, y) \to (-y, x) $$ This corresponds to a 90-degree rotation counterclockwise about the origin. Final answer: \((x, y) \to (-y, x)\)