Question: Find the coordinates of the image of $R(2, 1)$ after the translation $(x, y) \to (x, y - 5)$.
$(-3, -4)$
$(-2, 4)$
$(2, -4)$
$(-3, 1)$
1. **State the problem:** We need to find the new coordinates of point $R(2, 1)$ after applying the translation rule $(x, y) \to (x, y - 5)$.
2. **Understand the translation:** The translation rule means the $x$-coordinate remains the same, and the $y$-coordinate is decreased by $5$.
3. **Apply the translation:**
$$
(x, y) = (2, 1) \to (2, 1 - 5)
$$
4. **Calculate the new $y$-coordinate:**
$$
1 - 5 = -4
$$
5. **Write the new coordinates:**
$$
(2, -4)
$$
6. **Check the options:** The correct answer is $(2, -4)$.
**Final answer:** $(2, -4)$