Question

which sequence of transformations carries abcd onto efgh? a. reflection across the x - axis followed by reflection across the y - axis b. rotation of 90° clockwise about the origin followed by reflection across the x - axis c. translation of three units up followed by reflection across the y - axis d. reflection across the x - axis followed by rotation of 90° counterclockwise about the origin

Explanation:

Step1: Analyze Option A

• Reflection across the \(x -\)axis: \((x,y)\to(x, - y)\). For example, point \(A(2,-1)\) becomes \((2,1)\). Then reflection across the \(y -\)axis: \((x,y)\to(-x,y)\). So \((2,1)\) becomes \((- 2,1)\). Similarly, for \(B(2,-3)\to(2,3)\to(-2,3)\), \(C(6,-3)\to(6,3)\to(-6,3)\), \(D(6,-1)\to(6,1)\to(-6,1)\) which matches \(E(-1,1)\), \(F(-1,3)\), \(G(-6,3)\), \(H(-6,1)\) (after re - labeling the order of vertices as per the transformation of the rectangle).

Step2: Analyze Option B

• Rotation of \(90^{\circ}\) clockwise about the origin: \((x,y)\to(y,-x)\). For \(A(2,-1)\to(-1,-2)\), then reflection across the \(x -\)axis: \((-1,-2)\to(-1,2)\) which does not match the corresponding vertex of \(EFGH\).

Step3: Analyze Option C

• Translation of three units up: \((x,y)\to(x,y + 3)\). For \(A(2,-1)\to(2,2)\), then reflection across the \(y -\)axis: \((2,2)\to(-2,2)\) which does not match the corresponding vertex of \(EFGH\).

Step4: Analyze Option D

• Reflection across the \(x -\)axis: \((x,y)\to(x,-y)\). For \(A(2,-1)\to(2,1)\), then rotation of \(90^{\circ}\) counter - clockwise about the origin: \((x,y)\to(-y,x)\). So \((2,1)\to(-1,2)\) which does not match the corresponding vertex of \(EFGH\).

Answer:

A. Reflection across the \(x -\)axis followed by reflection across the \(y -\)axis