What is the image point of left parenthesis, minus, 4, comma, minus, 5, right parenthesis(−4,−5) after the transformation T, start subscript, 4, comma, 2, end subscript, circle, r, start subscript, y, equals, x, end subscriptT 
4,2
​
 ∘r 
y=x
​
 

To find the image point of \((-4, -5)\) after the transformation \( T_{4,2} \circ r_{y=x} \), we will follow these steps:

1. **Apply the reflection \( r_{y=x} \)**: The reflection across the line \( y = x \) swaps the \( x \) and \( y \) coordinates of the point.
   
   \[
   (-4, -5) \rightarrow (-5, -4)
   \]

2. **Apply the translation \( T_{4,2} \)**: The translation \( T_{4,2} \) adds 4 to the \( x \) coordinate and 2 to the \( y \) coordinate of the point.

   \[
   (-5, -4) \rightarrow (-5 + 4, -4 + 2) = (-1, -2)
   \]

Thus, the image point of \((-4, -5)\) after the transformation \( T_{4,2} \circ r_{y=x} \) is \(\boxed{(-1, -2)}\).

Would you like to go through any additional details or have more questions about this? Here are 5 related questions to consider:

1. What is the result if the reflection is across the \( y \)-axis instead?
2. How does the order of transformations affect the final result?
3. What is the geometric interpretation of \( T_{4,2} \circ r_{y=x} \)?
4. What happens if the point \((-4, -5)\) is first translated and then reflected?
5. How would the result change if the translation vector was \( (4, -2) \) instead?

**Tip:** Always perform transformations in the correct order, as it can significantly change the outcome.

What is the image point of left parenthesis, minus, 4, comma, minus, 5, right parenthesis(−4,−5) after the transformation T, start subscript, 4, comma, 2, end subscript, circle, r, start subscript, y, equals, x, end subscriptT4,2∘ry=x?

Discover how to find the image of the point (-4, -5) after performing a reflection across the line y = x followed by a translation by vector (4, 2). This problem involves key concepts in coordinate geometry and is suitable for students in Grades 9-11.

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