Let's break down the transformation step by step.

You are dealing with two transformations:

1. Dilation by a factor of $\frac{1}{3}$
   • Denoted as $D_{\frac{1}{3}}$, this transformation shrinks the point by a factor of $\frac{1}{3}$.
2. Reflection over the line $y = -x$
   • Denoted as $r_{y = -x}$, this reflects a point across the line $y = -x$.

Step 1: Apply the dilation $D_{\frac{1}{3}}$

The dilation $D_{\frac{1}{3}}$ scales both coordinates by $\frac{1}{3}$.

Given point $(6, 6)$, apply the dilation: $D_{\frac{1}{3}}(6, 6) = \left( \frac{1}{3} \cdot 6, \frac{1}{3} \cdot 6 \right) = (2, 2)$

Step 2: Apply the reflection $r_{y = -x}$

After dilation, we reflect the point $(2, 2)$ across the line $y = -x$. The reflection rule across $y = -x$ swaps and negates the coordinates: $r_{y = -x}(2, 2) = (-2, -2)$

Final Answer:

The image point of $(6, 6)$ after the transformation $r_{y = -x} \circ D_{\frac{1}{3}}$ is $(-2, -2)$.

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:

1. How do you apply a dilation transformation with a different factor?
2. What is the general formula for reflecting points over a line?
3. How do composite transformations work geometrically?
4. What would happen if the dilation factor was negative?
5. How would you describe the composition of two reflections?

Tip: In a reflection over $y = -x$, the coordinates of any point swap and change signs!