Let's break down the steps to solve this transformation problem:

1. Translation by $T\left(-1, 2\right)$: Each point $(x, y)$ will be translated by moving $x$ to $x - 1$ and $y$ to $y + 2$.

   • $P(2, -6)$ → $P_1(2 - 1, -6 + 2) = P_1(1, -4)$
   • $Q(-3, 5)$ → $Q_1(-3 - 1, 5 + 2) = Q_1(-4, 7)$
   • $R(-1, -4)$ → $R_1(-1 - 1, -4 + 2) = R_1(-2, -2)$

2. Reflection across the line $x = 2$: To reflect a point $(x, y)$ across the line $x = 2$, we apply the transformation: $x' = 2 + (2 - x) = 4 - x$

   • $P_1(1, -4)$ → $P_2(4 - 1, -4) = P_2(3, -4)$
   • $Q_1(-4, 7)$ → $Q_2(4 - (-4), 7) = Q_2(8, 7)$
   • $R_1(-2, -2)$ → $R_2(4 - (-2), -2) = R_2(6, -2)$

3. Rotation by $90^\circ$ counterclockwise around the origin $O(0, 0)$: The rotation formula for a $90^\circ$ counterclockwise rotation is: $x' = -y \quad \text{and} \quad y' = x$

   • $P_2(3, -4)$ → $P'(4, 3)$
   • $Q_2(8, 7)$ → $Q'(-7, 8)$
   • $R_2(6, -2)$ → $R'(2, 6)$

Now, comparing the transformed coordinates to the answer choices:

• The correct answer is D: $P'(8, 3), Q'(-3, -8), R'(6, 6)$.

Do you need any further details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you derive reflection formulas for lines other than $x = 2$?
2. What happens if you reflect a point over the $y$-axis instead?
3. How do rotations change when the angle is different from $90^\circ$?
4. What are the general transformation rules for scaling?
5. How can you determine the center of rotation for arbitrary points?

Tip: Always apply transformations step-by-step to avoid confusion and ensure accuracy!