To determine the transformation of the parent function \( f(x) = x^3 \) to the function \( g(x) = (x - 1)^3 - 2 \), we can analyze the modifications made to the function:
-
Horizontal Shift: The term \( (x - 1) \) indicates a shift of the graph 1 unit to the right. This is because replacing \( x \) with \( x - 1 \) translates the graph horizontally.
-
Vertical Shift: The \( -2 \) outside of the cubic function indicates a shift of the graph 2 units down. This means that every point on the graph of \( f(x) \) is lowered by 2 units.
Combining these transformations, we see that the function \( g(x) \) represents a shift of the parent function \( f(x) = x^3 \) 1 unit right and 2 units down.
So the correct option is:
1 unit right and 2 units down.