To determine the graph of the transformed function [tex]\( g(x) \)[/tex] from the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], we need to analyze the transformations step by step.
The given transformation is [tex]\( g(x) = f(x+2) - 4 \)[/tex].
1. Horizontal Shift:
- The expression [tex]\( f(x+2) \)[/tex] indicates a horizontal shift.
- The [tex]\( +2 \)[/tex] inside the function translates to a shift to the left by 2 units.
- Therefore, the graph of [tex]\( f(x) \)[/tex] is shifted left by 2 units.
2. Vertical Shift:
- The expression [tex]\( - 4 \)[/tex] outside the function indicates a vertical shift.
- The [tex]\( -4 \)[/tex] means the graph is shifted down by 4 units.
- Therefore, the graph of [tex]\( f(x) \)[/tex] is shifted down by 4 units.
Putting these transformations together:
- Start with the graph of the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
- Shift this graph 2 units to the left.
- Then, shift the graph 4 units downward.
The correct graph of [tex]\( g(x) = \sqrt[3]{x+2} - 4 \)[/tex] will show these combined transformations. By this reasoning, we find the graph that accurately represents these shifts among the given options.
