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]
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]
