Question

select the correct answer. which sequence is not equivalent to the others? a. a reflection across the y - axis, followed by a reflection across the x - axis, and then a 90° clockwise rotation about the origin b. a 90° clockwise rotation about the origin and then a 180° rotation about the origin c. a reflection across the x - axis, followed by a 90° counterclockwise rotation about the origin, and then a reflection across the x - axis d. a 90° counterclockwise rotation about the origin

Explanation:

Step1: Define transformation rules

Use coordinate transformations:

• Reflect over y-axis: $(x,y)\to(-x,y)$
• Reflect over x-axis: $(x,y)\to(x,-y)$
• 90° clockwise rotation (CW): $(x,y)\to(y,-x)$
• 180° rotation: $(x,y)\to(-x,-y)$
• 90° counterclockwise rotation (CCW): $(x,y)\to(-y,x)$

Step2: Compute Option A

Reflect over y-axis → reflect over x-axis → 90° CW rotation:

1. $(-x,y)$ → 2. $(-x,-y)$ → 3. $(-y,x)$

Step3: Compute Option B

90° CW rotation → 180° rotation:

1. $(y,-x)$ → 2. $(-y,x)$

Step4: Compute Option C

Reflect over x-axis → 90° CCW rotation → reflect over x-axis:

1. $(x,-y)$ → 2. $(y,x)$ → 3. $(y,-x)$

Step5: Compute Option D

90° CCW rotation:
$(x,y)\to(-y,x)$

Answer:

C. a reflection across the x-axis, followed by a 90° counterclockwise rotation about the origin, and then a reflection across the x-axis