1. **State the problem:** We need to find which sequence of transformations maps triangle ABC with vertices A(-6,1), B(-3,6), C(-2,2) to triangle A'' B'' C'' with vertices A''(4,-6), B''(6,-2), C''(2,-4). 2. **Analyze the given transformations:** - Translation moves points by adding/subtracting values to coordinates. - Reflection across the x-axis changes $(x,y)$ to $(x,-y)$. - Reflection across the y-axis changes $(x,y)$ to $(-x,y)$. 3. **Check each option:** **Option 1:** Translate 10 units right, then reflect across x-axis. - Translate right 10: $A(-6,1) \to (-6+10,1) = (4,1)$ - Reflect across x-axis: $(4,1) \to (4,-1)$ - But $A''$ is at $(4,-6)$, so this does not match. **Option 2:** Reflect across x-axis, then translate 9 units right. - Reflect across x-axis: $A(-6,1) \to (-6,-1)$ - Translate right 9: $(-6+9,-1) = (3,-1)$ - $A''$ is at $(4,-6)$, so no match. **Option 3:** Translate 7 units down, then 9 units right. - Translate down 7: $A(-6,1) \to (-6,1-7) = (-6,-6)$ - Translate right 9: $(-6+9,-6) = (3,-6)$ - $A''$ is at $(4,-6)$, so no match. **Option 4:** Reflect across y-axis, then translate 7 units down. - Reflect across y-axis: $A(-6,1) \to (6,1)$ - Translate down 7: $(6,1-7) = (6,-6)$ - $A''$ is at $(4,-6)$, so no match for point A. 4. **Check other points for option 1:** - For B(-3,6): translate right 10: $(7,6)$, reflect x-axis: $(7,-6)$ - $B''$ is at $(6,-2)$, no match. 5. **Check other points for option 2:** - B(-3,6): reflect x-axis: $(-3,-6)$, translate right 9: $(6,-6)$ - $B''$ is at $(6,-2)$, no match. 6. **Check other points for option 3:** - B(-3,6): down 7: $(-3,-1)$, right 9: $(6,-1)$ - $B''$ is at $(6,-2)$, close but no match. 7. **Check other points for option 4:** - B(-3,6): reflect y-axis: $(3,6)$, down 7: $(3,-1)$ - $B''$ is at $(6,-2)$, no match. 8. **Re-examine option 1 for point C(-2,2):** - Translate right 10: $(8,2)$ - Reflect x-axis: $(8,-2)$ - $C''$ is at $(2,-4)$, no match. 9. **Try to find a consistent transformation:** - From A to A'': $(-6,1) \to (4,-6)$ - The change in x: $4 - (-6) = 10$ - The change in y: $-6 - 1 = -7$ 10. **Check if reflecting across x-axis then translating 10 right and 7 down works:** - Reflect x-axis: $(-6,1) \to (-6,-1)$ - Translate right 10: $(-6+10,-1) = (4,-1)$ - Translate down 7: $(4,-1-7) = (4,-8)$ - $A''$ is at $(4,-6)$, no match. 11. **Check if reflecting across y-axis then translating 7 down and 9 right works:** - Reflect y-axis: $(-6,1) \to (6,1)$ - Translate down 7: $(6,-6)$ - Translate right 9: $(15,-6)$ - No match. 12. **Conclusion:** None of the given options exactly match the transformation from ABC to A''B''C'' based on the points. **Final answer:** None of the provided sequences of transformations correctly map triangle ABC to triangle A'' B'' C'' based on the given coordinates.