To analyze how the graph of \( g(x) = (x + 2)^3 - 7 \) compares to the parent function \( f(x) = x^3 \), we can identify the transformations involved.
-
Shift to the left/right: The expression \( (x + 2)^3 \) indicates a horizontal shift. In general, if we have \( (x - h)^3 \), the graph shifts \( h \) units to the right if \( h \) is positive and \( |h| \) units to the left if \( h \) is negative. In this case, \( (x + 2) \) means the graph is shifted 2 units to the left (since \( h = -2 \)).
-
Shift up/down: The \( -7 \) at the end of the function indicates a vertical shift. If we have \( f(x) - k \), where \( k \) is positive, the graph shifts \( k \) units down. Here, \( g(x) = (x + 2)^3 - 7 \) means the graph is shifted 7 units down.
Combining these two observations, we conclude that:
- The graph of \( g(x) \) is shifted 2 units to the left and 7 units down compared to the parent function \( f(x) = x^3 \).
Thus, the correct answer is:
g(x) is shifted 2 units to the left and 7 units down.