Let's begin by understanding the transformation rule [tex]\( R_{0,270^{\circ}} \)[/tex]. This rule indicates a rotation of 270 degrees counterclockwise about the origin.
For a point [tex]\((x, y)\)[/tex], rotating it 270 degrees counterclockwise about the origin means transforming it in such a way that its new coordinates [tex]\((x', y')\)[/tex] are given by:
[tex]\[ x' = y \][/tex]
[tex]\[ y' = -x \][/tex]
We need to apply this transformation to the coordinates of point [tex]\(S\)[/tex], which are [tex]\((-2, -4)\)[/tex].
Following the rules of transformation:
1. Assign [tex]\( x = -2 \)[/tex] and [tex]\( y = -4 \)[/tex].
2. Calculate the new x-coordinate [tex]\(x'\)[/tex]:
[tex]\[ x' = y = -4 \][/tex]
3. Calculate the new y-coordinate [tex]\(y'\)[/tex]:
[tex]\[ y' = -x = -(-2) = 2 \][/tex]
Therefore, the coordinates of [tex]\(S'\)[/tex] after a 270-degree counterclockwise rotation about the origin are [tex]\((-4, 2)\)[/tex].
The correct answer is:
[tex]\[
(-4, 2)
\][/tex]
