Question

4 which series of transformations will map rectangle abcd to rectangle abcd? a reflection in the y - axis and then the translation $(x + 4,y - 3)$ b the translation $(x + 2,y - 1)$ and then a reflection in the x - axis c reflection in the x - axis and then the translation $(x + 2,y + 1)$ d rotation of $180^{circ}$ about the origin and then the translation $(x + 4,y - 1)$

Explanation:

Step1: Analyze Option A

• Reflection in the \(y\) - axis: For a point \((x,y)\) reflected over the \(y\) - axis, the transformation is \((x,y)\to(-x,y)\).
• Then translation \((x + 4,y-3)\): After reflection, applying \((x,y)\to(x + 4,y - 3)\).
• Let's take a point, say \(A(1,1)\). After reflection over \(y\) - axis: \((-1,1)\). After translation: \((-1+4,1 - 3)=(3,-2)

eq A'\) (from the graph, assume coordinates based on grid).

Step2: Analyze Option B

• Translation \((x + 2,y-1)\): For a point \((x,y)\), it becomes \((x + 2,y-1)\).
• Then reflection in the \(x\) - axis: For a point \((x,y)\) reflected over the \(x\) - axis, the transformation is \((x,y)\to(x,-y)\).
• Take \(A(1,1)\). After translation: \((1+2,1 - 1)=(3,0)\). After reflection: \((3,0)\to(3,0)

eq A'\) (from the graph).

Step3: Analyze Option C

• Reflection in the \(x\) - axis: For a point \((x,y)\) reflected over the \(x\) - axis, the transformation is \((x,y)\to(x,-y)\).
• Then translation \((x + 2,y + 1)\): After reflection, applying \((x,y)\to(x + 2,y + 1)\).
• Take \(A(1,1)\). After reflection: \((1,-1)\). After translation: \((1+2,-1 + 1)=(3,0)=A'\) (assuming \(A'\) is at \((3,0)\) from the graph). Check another point, say \(B(5,1)\). After reflection: \((5,-1)\). After translation: \((5+2,-1 + 1)=(7,0)=B'\) (assuming \(B'\) is at \((7,0)\) from the graph).

Step4: Analyze Option D

• Rotation of \(180^{\circ}\) about the origin: For a point \((x,y)\) rotated \(180^{\circ}\) about the origin, the transformation is \((x,y)\to(-x,-y)\).
• Then translation \((x + 4,y-1)\).
• Take \(A(1,1)\). After rotation: \((-1,-1)\). After translation: \((-1+4,-1 - 1)=(3,-2)

eq A'\) (from the graph).

Answer:

C. Reflection in the \(x\) - axis and then the translation \((x + 2,y + 1)\)