1. **Problem statement:** Reflect points across the line $y = -x$ and then across the line $y = -1$. Given points $X(-3,-1)$, $Y(-1,1)$, and $Z(3,2)$, find the coordinates of $X''$, $Y''$, and $Z''$ after both reflections. 2. **Reflection across $y = -x$:** The reflection of a point $(a,b)$ across $y = -x$ is given by swapping and negating coordinates: $$ (a,b) \to (-b,-a) $$ 3. **Apply reflection across $y = -x$ to each point:** - $X(-3,-1) \to X' = (-(-1), -(-3)) = (1,3)$ - $Y(-1,1) \to Y' = ( -1, 1) \to (-1,-1)$ (correcting: $Y(-1,1) \to (-1,-1)$) - $Z(3,2) \to Z' = (-2,-3)$ 4. **Reflection across $y = -1$:** The line $y = -1$ is horizontal. Reflection across this line changes the $y$-coordinate as follows: $$ y' = -1 - (y - (-1)) = -1 - (y + 1) = -2 - y $$ The $x$-coordinate remains the same. 5. **Apply reflection across $y = -1$ to each $X'$, $Y'$, $Z'$:** - $X'(1,3) \to X'' = (1, -2 - 3) = (1, -5)$ - $Y'(-1,-1) \to Y'' = (-1, -2 - (-1)) = (-1, -1)$ - $Z'(-2,-3) \to Z'' = (-2, -2 - (-3)) = (-2, 1)$ 6. **Final answers:** - $X'' = (1, -5)$ - $Y'' = (-1, -1)$ - $Z'' = (-2, 1)$ These are the coordinates after reflecting first across $y = -x$ and then across $y = -1$.