Question

2 exam: semester 2 exam
ich sequence or transformations carnes abcd onto efgh
a. reflection across the y - axis followed by reflection across the x - axis
b. reflection across the x - axis followed by rotation of 90° counterclockwise about the origin
c. translation of seven units to the left followed by reflection across the x - axis
d. rotation of 90° clockwise about the origin followed by reflection across the x - axis

Explanation:

Step1: Analyze Option A

• Reflection across \(y\) - axis:
• For a point \((x,y)\) on \(ABCD\), after reflection across \(y\) - axis, it becomes \((-x,y)\).
• Then reflection across \(x\) - axis: For a point \((-x,y)\), after reflection across \(x\) - axis, it becomes \((-x,-y)\).
• Let's take a point \(A(2,-1)\) on \(ABCD\). After reflection across \(y\) - axis, it becomes \((-2,-1)\), then after reflection across \(x\) - axis, it becomes \((-2,1)\) which is the position of \(F\) (assuming proper correspondence).
• Take another point \(B(2,-3)\) on \(ABCD\). After reflection across \(y\) - axis, it becomes \((-2,-3)\), then after reflection across \(x\) - axis, it becomes \((-2,3)\) which is the position of \(E\) (assuming proper correspondence).

Step2: Analyze Option B

• Reflection across \(x\) - axis:
• For a point \((x,y)\) on \(ABCD\), after reflection across \(x\) - axis, it becomes \((x,-y)\).
• Then rotation of \(90^{\circ}\) counter - clockwise about the origin:
• The rotation formula for a point \((x,-y)\) is \((y,x)\).
• Take point \(A(2,-1)\), after reflection across \(x\) - axis, it becomes \((2,1)\), then after rotation of \(90^{\circ}\) counter - clockwise about the origin, it becomes \((1,2)\) which does not match the position of any point on \(EFGH\).

Step3: Analyze Option C

• Translation of seven units to the left:
• For a point \((x,y)\) on \(ABCD\), after translation of seven units to the left, it becomes \((x - 7,y)\).
• Then reflection across \(x\) - axis: For a point \((x - 7,y)\), after reflection across \(x\) - axis, it becomes \((x - 7,-y)\).
• Take point \(A(2,-1)\), after translation of seven units to the left, it becomes \((-5,-1)\), then after reflection across \(x\) - axis, it becomes \((-5,1)\) which does not match the position of any point on \(EFGH\).

Step4: Analyze Option D

• Rotation of \(90^{\circ}\) clockwise about the origin:
• The rotation formula for a point \((x,y)\) is \((y,-x)\).
• Then reflection across \(x\) - axis: For a point \((y,-x)\), after reflection across \(x\) - axis, it becomes \((y,x)\).
• Take point \(A(2,-1)\), after rotation of \(90^{\circ}\) clockwise about the origin, it becomes \((-1,-2)\), then after reflection across \(x\) - axis, it becomes \((-1,2)\) which does not match the position of any point on \(EFGH\).

Answer:

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