To determine the image of the point $(-6, -4)$ under the transformation $T_{-4, -1} \circ r_{y=-x}$, we need to understand that this is a composition of two transformations:

1. Reflection across the line $y = -x$ ($r_{y=-x}$)
2. Translation by the vector $(-4, -1)$ ($T_{-4, -1}$)

Step 1: Apply the Reflection $r_{y=-x}$

Reflecting a point $(x, y)$ across the line $y = -x$ involves swapping the coordinates and changing their signs: $r_{y=-x}(x, y) = (-y, -x)$ For the point $(-6, -4)$, this becomes: $r_{y=-x}(-6, -4) = (4, 6)$

Step 2: Apply the Translation $T_{-4, -1}$

Now we apply the translation $T_{-4, -1}$ to the point $(4, 6)$. A translation by $(-4, -1)$ means subtracting 4 from the $x$-coordinate and 1 from the $y$-coordinate: $T_{-4, -1}(4, 6) = (4 - 4, 6 - 1) = (0, 5)$

Final Answer

The image of the point $(-6, -4)$ under the transformation $T_{-4, -1} \circ r_{y=-x}$ is: $(0, 5)$

Would you like more details or have any questions about this solution?

Here are some related questions for further practice:

1. What is the effect of reflecting a point over the line $y = x$ instead of $y = -x$?
2. How would you find the image of a point under a rotation followed by a reflection?
3. What is the result if we only apply the translation $T_{-4, -1}$ without the reflection?
4. How do you determine the inverse of the transformation $T_{-4, -1} \circ r_{y=-x}$?
5. What is the result of applying $r_{y=-x} \circ T_{-4, -1}$ instead of $T_{-4, -1} \circ r_{y=-x}$?

Tip: When working with compositions of transformations, remember to apply them in the specified order, starting from the innermost transformation and working outward.