1. **Problem Statement:** We have a triangle with vertices at points $P(2, -1)$, $Q(2, -3)$, and $R(4, -3)$. We want to find the image of this triangle after a translation along the vector $\langle -4, 1 \rangle$. 2. **Transformation Rule:** A translation by vector $\langle a, b \rangle$ moves every point $(x, y)$ to $(x + a, y + b)$. Here, $a = -4$ and $b = 1$, so the rule is: $$ (x, y) \to (x - 4, y + 1) $$ 3. **Apply the Translation to Each Vertex:** - For $P(2, -1)$: $$ (2, -1) \to (2 - 4, -1 + 1) = (-2, 0) $$ - For $Q(2, -3)$: $$ (2, -3) \to (2 - 4, -3 + 1) = (-2, -2) $$ - For $R(4, -3)$: $$ (4, -3) \to (4 - 4, -3 + 1) = (0, -2) $$ 4. **Summary:** - Preimage vertices: $P(2, -1)$, $Q(2, -3)$, $R(4, -3)$ - Image vertices after translation: $P'(-2, 0)$, $Q'(-2, -2)$, $R'(0, -2)$ 5. **Explanation:** Translation moves every point the same amount without changing the shape or size of the figure. Here, each point moves 4 units left and 1 unit up. **Final answer:** $$\text{Transformation rule: } (x, y) \to (x - 4, y + 1)$$ $$P'( -2, 0),\quad Q'(-2, -2),\quad R'(0, -2)$$