1. **State the problem:** We have a parallelogram with preimage vertices at $(1,5)$, $(3,3)$, $(3,7)$, and $(5,5)$, and its image vertices at $(-5,3)$, $(-3,1)$, $(-3,5)$, and $(-1,3)$. We need to find the translation vector that maps the preimage to the image. 2. **Formula and concept:** A translation moves every point of a figure the same distance in the same direction. The mapping notation for a translation is: $$ (x,y) \to (x + a, y + b) $$ where $(a,b)$ is the translation vector. 3. **Find the translation vector:** To find $(a,b)$, subtract the coordinates of a preimage vertex from its corresponding image vertex. Using the first vertex: $$ a = -5 - 1 = -6 $$ $$ b = 3 - 5 = -2 $$ 4. **Check with another vertex:** Using the second vertex: $$ a = -3 - 3 = -6 $$ $$ b = 1 - 3 = -2 $$ The translation vector is consistent. 5. **Write the mapping notation:** $$ (x,y) \to (x - 6, y - 2) $$ 6. **Interpretation:** The parallelogram was translated 6 units to the left and 2 units down. **Final answer:** The translation is given by the mapping $$ (x,y) \to (x - 6, y - 2) $$ which means move every point 6 units left and 2 units down.