Question

which pair of transformations is the same as a reflection across the y - axis?
a a rotation of 90° counter - clockwise and a reflection across the x - axis
b a rotation of 90° clockwise and a reflection across the y - axis
c a rotation of 180° and a reflection across the x - axis
d a rotation of 180° and a reflection across the y - axis

Explanation:

Step1: Analyze each option

Let's assume a point \((x,y)\) in the coordinate plane.

• Option A:
• Rotation of \(90^{\circ}\) counter - clockwise: \((x,y)\to(-y,x)\)
• Reflection across the \(x\) - axis: \((-y,x)\to(-y,-x)\)
• Option B:
• Rotation of \(90^{\circ}\) clockwise: \((x,y)\to(y,-x)\)
• Reflection across the \(y\) - axis: \((y,-x)\to(-y,-x)\)
• Option C:
• Rotation of \(180^{\circ}\): \((x,y)\to(-x,-y)\)
• Reflection across the \(x\) - axis: \((-x,-y)\to(-x,y)\)
• Option D:
• Rotation of \(180^{\circ}\): \((x,y)\to(-x,-y)\)
• Reflection across the \(y\) - axis: \((-x,-y)\to(x,-y)\)

Step2: Compare with reflection across the \(y\) - axis

Reflection across the \(y\) - axis: \((x,y)\to(-x,y)\)

Answer:

C. a rotation of \(180^{\circ}\) and a reflection across the \(x\) - axis