To find the corresponding point \( A' \) after a dilation of point \( A (-3, 1) \) with a center of dilation at \( (-2, -1) \) and a magnitude of 2, we can use the formula for dilation:
\[ A' = (1 - k) O + k A \]
Where:
- \( O \) is the center of dilation
- \( k \) is the scale factor (in this case, \( k = 2 \))
- \( A \) is the original point
Given:
- \( O = (-2, -1) \)
- \( A = (-3, 1) \)
- \( k = 2 \)
Now we can calculate the coordinates of \( A' \):
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Calculate \( A - O \): \[ A - O = (-3, 1) - (-2, -1) = (-3 + 2, 1 + 1) = (-1, 2) \]
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Scale this vector by \( k = 2 \): \[ k(A - O) = 2 \cdot (-1, 2) = (-2, 4) \]
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Add this to the center of dilation \( O \): \[ A' = O + k(A - O) = (-2, -1) + (-2, 4) = (-2 - 2, -1 + 4) = (-4, 3) \]
Thus, the corresponding ordered pair for point \( A \) after the dilation is:
\[ \boxed{(-4, 3)} \]