To solve the problem, let us first understand the translation $T$ based on the given information:

• The point $Q(-4, 2)$ is mapped to $Q'(-1, 6)$.
• Translation $T$ applies the same rule to map the point $R(3, -2)$.

Step 1: Determine the translation vector

The translation vector can be found by calculating the difference between the coordinates of $Q$ and $Q'$:

$\text{Translation vector} = (x_{\text{new}} - x_{\text{old}}, y_{\text{new}} - y_{\text{old}})$

$\text{Translation vector} = (-1 - (-4), 6 - 2) = (3, 4)$

Step 2: Apply the translation to $R(3, -2)$

Add the translation vector $(3, 4)$ to the coordinates of $R(3, -2)$:

$R'(x, y) = (x + 3, y + 4)$

$R'(3 + 3, -2 + 4) = (6, 2)$

Final Answer:

The translated point is $R'(6, 2)$. Thus, the correct option is:

D. $R'(6, 2)$

Would you like additional clarification? Here are 5 related questions to expand your understanding:

1. What happens if the translation vector is reversed?
2. How would you apply this translation to other points in the same system?
3. How can translations be represented in matrix form?
4. What is the difference between translation and reflection in geometry?
5. Can a translation change the size or shape of a figure?

Tip: Translation only shifts a point or object without altering its size, shape, or orientation.