Let's break down the sequence of transformations applied to the point [tex]\((x, y)\)[/tex] into detailed steps:
### Step 1: Rotate 180 degrees clockwise about the origin
When we rotate a point [tex]\((x, y)\)[/tex] 180 degrees clockwise about the origin, the new coordinates of the point are given by [tex]\((-x, -y)\)[/tex].
### Step 2: Translate 6 units to the left
Next, we translate the point obtained from step 1 six units to the left. If we have a point [tex]\((x', y')\)[/tex] and we translate it 6 units to the left, the transformation can be represented by:
[tex]\[
(x', y') \rightarrow (x' - 6, y')
\][/tex]
Applying this to the coordinates from step 1 [tex]\((-x, -y)\)[/tex]:
[tex]\((-x, -y) \rightarrow (-x - 6, -y)\)[/tex]
### Step 3: Reflect across the line [tex]\(y=x\)[/tex]
Finally, reflecting a point [tex]\((x'', y'')\)[/tex] across the line [tex]\(y=x\)[/tex] interchanges its coordinates, resulting in the transformation:
[tex]\[
(x'', y'') \rightarrow (y'', x'')
\][/tex]
Applying this to the coordinates from step 2 [tex]\((-x - 6, -y)\)[/tex]:
[tex]\((-x - 6, -y) \rightarrow (-y, -x - 6)\)[/tex]
Combining all these transformations, we can represent the function [tex]\(S\)[/tex] that maps an initial point [tex]\((x, y)\)[/tex] through these transformations:
1. Rotate 180 degrees clockwise: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
2. Translate 6 units to the left: [tex]\((-x, -y) \rightarrow (-x - 6, -y)\)[/tex]
3. Reflect across the line [tex]\(y=x\)[/tex]: [tex]\((-x - 6, -y) \rightarrow (-y, -x - 6)\)[/tex]
Thus, the function [tex]\(S\)[/tex] which represents the sequence of transformations applied to the point [tex]\((x, y)\)[/tex] is:
[tex]\[
S(x, y) = (-y, -x - 6)
\][/tex]
To summarize, the transformations are as follows:
[tex]\[
(x, y) \rightarrow (-x, -y) \rightarrow (-x - 6, -y) \rightarrow (-y, -x - 6)
\][/tex]
So the final coordinates of the point after all transformations are [tex]\((-y, -x - 6)\)[/tex].
