Answered

Which rule describes a composition of transformations that maps pre-image PQRS to image P"Q"R"S"?

A. \(R_{0,270^{\circ}} \circ T_{-2,0}(x, y)\)

B. \(T_{-2,0} \circ R_{0,270^{\circ}}(x, y)\)

C. \(R_{0,270^{\circ}} \circ r_{y-\text{axis}}(x, y)\)

D. [tex]\(r_{y-\text{axis}} \circ R_{0,270^{\circ}}(x, y)\)[/tex]



Answer :

To determine which rule describes the correct composition of transformations that maps a pre-image PQRS to an image P"Q"R"S", we need to consider a sequence of transformations and how they are applied.

Let's break down each of the options:

1. \( R_{0, 270^{\circ}} \circ T_{-2,0}(x, y) \)

This option represents a rotation of \( 270^{\circ} \) counterclockwise about the origin, followed by a translation of \(-2\) units along the x-axis.

2. \( T_{-2,0} \circ R_{0,270^{\circ}}(x, y) \)

This option represents a translation of \(-2\) units along the x-axis, followed by a rotation of \( 270^{\circ} \) counterclockwise about the origin.

3. \( R_{0, 270^{\circ}} \circ r_{y-\operatorname{axis}}(x, y) \)

This option represents a rotation of \( 270^{\circ} \) counterclockwise about the origin, followed by a reflection over the y-axis.

4. \( r_{y \text{-axis}} \circ R_{0,270^{\circ}}(x, y) \)

This option represents a reflection over the y-axis, followed by a rotation of \( 270^{\circ} \) counterclockwise about the origin.

To achieve the desired mapping, PQRS must first be rotated \( 270^{\circ} \) counterclockwise about the origin. After rotating, each point of PQRS must be translated \(-2\) units along the x-axis.

The correct operation involves performing the rotation first and then the translation. This corresponds accurately to the composition described in the first option:

[tex]\[ R_{0,270^{\circ}} \circ T_{-2,0}(x, y) \][/tex]

Therefore, the correct option is:

1. [tex]\( R_{0, 270^{\circ}} \circ T_{-2,0}(x, y) \)[/tex]