To find the image point of $(-6, -8)$ after the transformation $R_{90^\circ} \circ r_{y=-x}$, we need to break this down step by step.

Step 1: Reflection across the line $y = -x$

The reflection of a point $(x, y)$ across the line $y = -x$ results in the point $(-y, -x)$.

For the point $(-6, -8)$:

• Reflecting $(-6, -8)$ across $y = -x$ gives: $(-(-8), -(-6)) = (8, 6)$

Step 2: Rotation by $90^\circ$ counterclockwise

The point $(x, y)$ rotated $90^\circ$ counterclockwise about the origin becomes $(-y, x)$.

For the point $(8, 6)$:

• Rotating $(8, 6)$ $90^\circ$ counterclockwise gives: $(-6, 8)$

Final Answer:

The image point of $(-6, -8)$ after the transformation $R_{90^\circ} \circ r_{y=-x}$ is $(-6, 8)$.

Would you like details on any part of this process, or do you have any questions?

Here are 5 related questions you might find interesting:

1. What is the result of applying only the reflection $r_{y=-x}$ to a general point $(x, y)$?
2. How does the result differ if we rotate $(x, y)$ by $90^\circ$ clockwise instead of counterclockwise?
3. What is the effect of applying the rotation before the reflection?
4. How does the transformation $R_{180^\circ} \circ r_{y=-x}$ affect a general point $(x, y)$?
5. Can we represent the transformation $R_{90^\circ} \circ r_{y=-x}$ using matrix multiplication?

Tip: Visualizing transformations using sketches or coordinate grids can help you understand the steps more clearly.