To find the new position of point \( A(-6, -6) \) after the specified transformations, we can break it down into two steps: the translation and the reflection.
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Translation:
- Moving 2 units to the right means we add 2 to the x-coordinate.
- Moving 1 unit up means we add 1 to the y-coordinate.
Starting with point \( A(-6, -6) \): \[ x' = -6 + 2 = -4 \] \[ y' = -6 + 1 = -5 \] After the translation, point \( A \) moves to \( A'(-4, -5) \).
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Reflection over the x-axis:
- When reflecting a point over the x-axis, the x-coordinate remains the same, while the y-coordinate changes sign.
Starting with point \( A'(-4, -5) \): \[ x'' = -4 \] \[ y'' = -(-5) = 5 \] After the reflection, point \( A \) moves to \( A''(-4, 5) \).
Thus, after the translation and reflection, the final position of point \( A \) is: \[ \boxed{(-4, 5)} \]