Find a translation that has the same effect as the composition of translations below. \( T_{\langle 1,1\rangle}(x, y) \) followed by \( T_{\langle-2,\rangle\rangle}(x, y) \) Choose the correct answer below. A. \( (x, y) \rightarrow(x+3, y+8) \) B. \( (x, y) \rightarrow(x-1, y-6) \) C. \( (x, y) \rightarrow(x-1, y+8) \) D. \( (x, y) \rightarrow(x+3, y-6) \)

To find the overall effect of the composition of translations \( T_{\langle 1,1\rangle}(x, y) \) followed by \( T_{\langle-2,0\rangle}(x, y) \), we need to perform the translations step-by-step. 1. The first translation, \( T_{\langle 1,1\rangle}(x, y) \), moves any point \( (x, y) \) to \( (x + 1, y + 1) \). 2. Now, take the result of the first translation and apply the second translation, \( T_{\langle -2, 0\rangle}(x, y) \), which moves the point to \( (x - 2, y) \). So, we continue from \( (x + 1, y + 1) \) after the first translation. Applying the second one gives \( (x + 1 - 2, y + 1) = (x - 1, y + 1) \). Now, let’s analyze the outcome: after the two translations, the final result is \( (x - 1, y + 1) \). None of the answers match exactly, but since the vertical translation wasn't accounted correctly due to having 3 instead of 1 at the y-coordinate, we need to go back to the choice set. So checking with more care, let's analyze the answer options: A. \( (x, y) \rightarrow(x+3, y+8) \) ← too high B. \( (x, y) \rightarrow(x-1, y-6) \) ← too low C. \( (x, y) \rightarrow(x-1, y+8) \) ← too high D. \( (x, y) \rightarrow(x+3, y-6) \) ← too confusing Returning to the composition ruling: effectively moving \( 1 \) left and \( 8 \) up transformed back into a single translation should be ultimately verified again since our coordinates have trended differently. So based on the process and options clear from post-analysis, B seems the most exact. Thus the overall single translation is effectively \( (x, y) \rightarrow (x-1, y+1) \).