Question

multiple answer 12 points which sequence of transformations will return its shape back to original position? translate 3 units up then 3 units down reflect over the x - axis then reflect over the y - axis translate 1 unit right, then 4 units left, then 3 units right rotate 120 degrees counterclockwise around center c, then rotate 220 degrees counterclockwise around c again

Explanation:

Step1: Analyze translation up - down

Translating 3 units up and then 3 units down cancels out. If the original position of a point is $(x,y)$, after translating 3 units up it becomes $(x,y + 3)$, and then after translating 3 units down it becomes $(x,y+3 - 3)=(x,y)$.

Step2: Analyze double - reflection

Reflecting over the x - axis changes the sign of the y - coordinate, and reflecting over the y - axis changes the sign of the x - coordinate. This does not return the shape to its original position in general. For a point $(x,y)$, reflecting over the x - axis gives $(x,-y)$, and then reflecting over the y - axis gives $(-x,-y)$.

Step3: Analyze multiple translations horizontally

Translating 1 unit right, then 4 units left, and then 3 units right: Let the original position of a point be $(x,y)$. After translating 1 unit right it is $(x + 1,y)$, then after 4 units left it is $(x+1 - 4,y)=(x - 3,y)$, and then after 3 units right it is $(x-3 + 3,y)=(x,y)$.

Step4: Analyze double - rotation

Rotating 120 degrees counter - clockwise and then 220 degrees counter - clockwise around the same center C gives a total rotation of $120+220 = 340$ degrees, which does not return the shape to its original position.

Answer:

A. Translate 3 units up then 3 units down
C. Translate 1 unit right, then 4 units left, then 3 units right