Question

5
which sequence of transformations will result in an image that maps onto itself?
a. reflect over the y-axis, and then reflect over the x-axis.
b. reflect over the y-axis, and then reflect again over the y-axis.
c. rotate 180 degrees counterclockwise about the origin, and then reflect across the x-axis.
d. rotate 180 degrees counterclockwise about the origin, and then reflect across the y-axis.
reset next

Explanation:

Step1: Analyze Option A

Reflecting over the \( y \)-axis changes a point \((x,y)\) to \((-x,y)\). Then reflecting over the \( x \)-axis changes \((-x,y)\) to \((-x,-y)\), which is not the original point (unless \( x = 0,y = 0 \)), so this does not map the image onto itself.

Step2: Analyze Option B

Reflecting over the \( y \)-axis changes \((x,y)\) to \((-x,y)\). Reflecting again over the \( y \)-axis changes \((-x,y)\) back to \((x,y)\), which is the original point. So this sequence maps the image onto itself.

Step3: Analyze Option C

Rotating \( 180^\circ \) counterclockwise about the origin changes \((x,y)\) to \((-x,-y)\). Then reflecting across the \( x \)-axis changes \((-x,-y)\) to \((-x,y)\), which is not the original point, so this does not map the image onto itself.

Step4: Analyze Option D

Rotating \( 180^\circ \) counterclockwise about the origin changes \((x,y)\) to \((-x,-y)\). Then reflecting across the \( y \)-axis changes \((-x,-y)\) to \((x,-y)\), which is not the original point, so this does not map the image onto itself.

Answer:

B. Reflect over the \( y \)-axis, and then reflect again over the \( y \)-axis.