Question

Which of the following compositions of transformations will always produce the same image, regardless of the order in which the transformations are performed?

A. translation and reflection
B. reflection and rotation
C. translation and rotation
D. All compositions are affected by the order in which they are performed

Answer 1

The correct answer is:

D) All compositions are affected by the order in which they are performed .

Step-by-step explanation:

If you translate a figure and rotate it, this will have different coordinates than rotating it first and translating it second. For example, if we translate a figure using <3, 2> and then rotate it 90 degrees clockwise about the origin:

Each point (x, y) will be mapped first to (x+3, y+2) from the translation.

The rotation will then change every point (x, y) to (-y, x); this makes

(x+3, y+2)→(-(y+2), x+3) = (-y-2, x+3).

If we rotated first, (x, y) would become (y, -x). Then the translation would make it (y+3, -x+2).

If we reflected through the y-axis and rotated 90 degrees clockwise:

After the reflection, (x, y)→(-x, y).

After the rotation, (-x, y)→(y, x).

If we rotate first, (x, y)→(y, -x).

After the reflection, (y, -x)→(-y, -x).

If we translated by <3, 2> and reflected through the y-axis:

After the translation, (x, y)→(x+3, y+2).

After the reflection, (x+3, y+2)→(-(x+3), y+2) = (-x-3, y+2).

If we reflected first, (x, y)→(-x, y).

After the translation, (-x, y)→(-x+3, y+2).

Each of these had a different outcome based on the order the transformations were performed in.

Answer 2

"reflection and rotation" is the one compositions of transformations among the following choices given in the question that will always produce the same image, regardless of the order in which the transformations are performed. The correct option among all the options that are given in the question is the second option or option "B".