To analyze the transformation of the parent function \( f(x) = x^3 \) to the function \( g(x) = (x-1)^3 - 2 \), we need to understand how shifts work.
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Horizontal Shift: The expression \( (x - 1) \) indicates a horizontal shift. Specifically, \( (x - 1) \) shifts the graph 1 unit to the right.
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Vertical Shift: The expression \(-2\) outside the cubic function indicates a vertical shift. Specifically, subtracting 2 shifts the graph down by 2 units.
Combining these two shifts, we can conclude that the function \( g(x) = (x-1)^3 - 2 \) is the result of a shift of \( f(x) = x^3 \) 1 unit right and 2 units down.
So, the correct answer is: 1 unit right and 2 units down.