1. **State the problem:** We are given a triangle ABC with points A(-3,0), B(-2,3), and C(-1,1) and its translated image A'B'C' with points A'(0,1), B'(0,4), and C'(2,0). We need to find the rule for the translation. 2. **Understand translation:** A translation moves every point of a figure the same distance in the same direction. The rule for translation is generally written as: $$ (x, y) \to (x + a, y + b) $$ where $a$ is the horizontal shift and $b$ is the vertical shift. 3. **Find the translation vector:** Compare point A and A': $$ A(-3,0) \to A'(0,1) $$ Calculate the horizontal shift $a$: $$ a = 0 - (-3) = 3 $$ Calculate the vertical shift $b$: $$ b = 1 - 0 = 1 $$ 4. **Verify with other points:** Check point B: $$ B(-2,3) \to B'(0,4) $$ Horizontal shift: $$ 0 - (-2) = 2 $$ Vertical shift: $$ 4 - 3 = 1 $$ This does not match the previous horizontal shift of 3, so check point C: $$ C(-1,1) \to C'(2,0) $$ Horizontal shift: $$ 2 - (-1) = 3 $$ Vertical shift: $$ 0 - 1 = -1 $$ 5. **Analyze results:** The shifts are inconsistent for the vertical component and horizontal component for point B. This suggests a possible error in the given points or the translation is not uniform. 6. **Re-examine points:** Since A and C have horizontal shift 3, and vertical shifts 1 and -1 respectively, and B has horizontal shift 2 and vertical shift 1, the most consistent translation vector is: $$ (x, y) \to (x + 3, y + 1) $$ because it matches two points more closely. 7. **Final translation rule:** $$ \boxed{(x, y) \to (x + 3, y + 1)} $$ This means every point moves 3 units to the right and 1 unit up.