Question What is the image point of $$ (2,-4) $$ after the transformation $$ r_{y=x} \circ T_{0,0} $$ ?

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1. **Translation $$ T_{0,0} $$:**

The translation $$ T_{0,0} $$ moves any point by the vector $$ (0,0) $$. This means it leaves the point unchanged. Therefore,
   
   $$
   T_{0,0}(2,-4) = (2,-4)
   $$

2. **Reflection $$ r_{y=x} $$:**

The reflection $$ r_{y=x} $$ across the line $$ y=x $$ swaps the $$ x $$- and $$ y $$-coordinates of a point. Thus, for any point $$ (a,b) $$,
   
   $$
   r_{y=x}(a, b) = (b, a)
   $$
   
   Applying this to the point $$ (2,-4) $$:
   
   $$
   r_{y=x}(2, -4) = (-4, 2)
   $$

3. **Combined Transformation $$ r_{y=x} \circ T_{0,0} $$:**

Since $$ T_{0,0} $$ does not change the point, the combined transformation is simply the reflection:
   
   $$
   \left(r_{y=x} \circ T_{0,0}\right)(2,-4) = r_{y=x}(2,-4) = (-4,2)
   $$

Thus, the image point is $$ (-4,2) $$.
The image point after the transformation is $$ (-4, 2) $$.

Question What is the image point of \( (2,-4) \) after the transformation \( r_{y=x} \circ T_{0,0} \) ?

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