Answer :
Let's break down the composition of transformations given in the question to understand which rule correctly maps the pre-image \( PQRS \) to the image \( P'Q'R'S'' \).
### Transformations
1. Rotation \( R_{0, 270^\circ} \):
- This denotes a rotation of \( 270^\circ \) (counterclockwise) about the origin.
- Generally, rotating \((x, y)\) by \( 270^\circ \) counterclockwise about the origin results in \((y, -x)\).
2. Translation \( T_{-2,0} \):
- This denotes a translation by \(-2\) units along the x-axis.
- Translating \((x, y)\) by \(-2\) units along the x-axis changes the coordinates to \((x-2, y)\).
3. Reflection \( r_{y-2,xis(i)} \):
- Reflection across a line. However, since the parameters seem a bit off, for simplicity, consider typical reflection behaviors.
### Composition Rules
- When composing transformations, the order in which you apply them is crucial. \( A \circ B \) means you apply \( B \) first and then \( A \).
#### Let's analyze each option with some hypothetical points:
1. \( R_{0, 270^\circ} \circ T_{-2,0}(x, y) \):
- First translate \((x, y)\) to \((x-2, y)\).
- Then apply the \( 270^\circ \) rotation on \((x-2, y)\):
- Resulting in \((y, -(x-2)) = (y, -x + 2)\).
2. \( T_{-2,0} \circ R_{0, 270^\circ}(x, y) \):
- First apply the \( 270^\circ \) rotation on \((x, y)\):
- Rotation gives \((y, -x)\).
- Then translate \((y, -x)\) by \(-2\) units along the x-axis:
- Resulting in \((y-2, -x)\).
3. \( R_{0, 270^\circ} \circ r_{y-2 \cdot i \cdot s}(x, y) \):
- First reflect \((x, y)\) (parameter issue, assuming some reflection).
- Then apply the \( 270^\circ \) rotation.
4. \( r_{y-2 \cdot x \cdot i \cdot s} \circ R_{0, 270^\circ}(x, y) \):
- First apply the \( 270^\circ \) rotation on \((x, y)\):
- Giving \((y, -x)\).
- Then reflect (parameter issue, assume some reflection).
### Solution
From the detailed transformations and composition rules:
- The correct way to read and perform the transformations is: First rotation followed by translation or vice versa.
Thus, Option 2:
\( T_{-2,0} \circ R_{0, 270^\circ}(x, y) \) describes the correct composition:
First, apply the [tex]\( 270^\circ \)[/tex] rotation and then translate the result by [tex]\(-2\)[/tex] units along the x-axis.
### Transformations
1. Rotation \( R_{0, 270^\circ} \):
- This denotes a rotation of \( 270^\circ \) (counterclockwise) about the origin.
- Generally, rotating \((x, y)\) by \( 270^\circ \) counterclockwise about the origin results in \((y, -x)\).
2. Translation \( T_{-2,0} \):
- This denotes a translation by \(-2\) units along the x-axis.
- Translating \((x, y)\) by \(-2\) units along the x-axis changes the coordinates to \((x-2, y)\).
3. Reflection \( r_{y-2,xis(i)} \):
- Reflection across a line. However, since the parameters seem a bit off, for simplicity, consider typical reflection behaviors.
### Composition Rules
- When composing transformations, the order in which you apply them is crucial. \( A \circ B \) means you apply \( B \) first and then \( A \).
#### Let's analyze each option with some hypothetical points:
1. \( R_{0, 270^\circ} \circ T_{-2,0}(x, y) \):
- First translate \((x, y)\) to \((x-2, y)\).
- Then apply the \( 270^\circ \) rotation on \((x-2, y)\):
- Resulting in \((y, -(x-2)) = (y, -x + 2)\).
2. \( T_{-2,0} \circ R_{0, 270^\circ}(x, y) \):
- First apply the \( 270^\circ \) rotation on \((x, y)\):
- Rotation gives \((y, -x)\).
- Then translate \((y, -x)\) by \(-2\) units along the x-axis:
- Resulting in \((y-2, -x)\).
3. \( R_{0, 270^\circ} \circ r_{y-2 \cdot i \cdot s}(x, y) \):
- First reflect \((x, y)\) (parameter issue, assuming some reflection).
- Then apply the \( 270^\circ \) rotation.
4. \( r_{y-2 \cdot x \cdot i \cdot s} \circ R_{0, 270^\circ}(x, y) \):
- First apply the \( 270^\circ \) rotation on \((x, y)\):
- Giving \((y, -x)\).
- Then reflect (parameter issue, assume some reflection).
### Solution
From the detailed transformations and composition rules:
- The correct way to read and perform the transformations is: First rotation followed by translation or vice versa.
Thus, Option 2:
\( T_{-2,0} \circ R_{0, 270^\circ}(x, y) \) describes the correct composition:
First, apply the [tex]\( 270^\circ \)[/tex] rotation and then translate the result by [tex]\(-2\)[/tex] units along the x-axis.
