To determine the graph of the transformed function [tex]\( g(x) = \sqrt[3]{x + 2} - 4 \)[/tex], we need to understand how the transformations apply to the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
Step-by-Step Solution:
1. Identify the Parent Function:
The parent function is [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
2. Understand the Transformation:
The function [tex]\( g(x) = \sqrt[3]{x + 2} - 4 \)[/tex] is a transformation of the parent function [tex]\( f(x) \)[/tex].
3. Horizontal Translation:
The term [tex]\( \sqrt[3]{x + 2} \)[/tex] inside the cube root function indicates a horizontal shift. Specifically, [tex]\( x + 2 \)[/tex] means the graph of [tex]\( f(x) \)[/tex] is shifted left by 2 units. In general, a horizontal shift is given by [tex]\( x - h \)[/tex], which moves the graph right by [tex]\( h \)[/tex] units if [tex]\( h \)[/tex] is positive and left if [tex]\( h \)[/tex] is negative.
4. Vertical Translation:
The term [tex]\( - 4 \)[/tex] outside the cube root function indicates a vertical shift. Specifically, subtracting 4 means the graph is shifted down by 4 units. In general, [tex]\( f(x) + k \)[/tex] moves the graph up by [tex]\( k \)[/tex] units if [tex]\( k \)[/tex] is positive and down if [tex]\( k \)[/tex] is negative.
Combining the Transformations:
- The graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is shifted left by 2 units to create [tex]\( \sqrt[3]{x + 2} \)[/tex].
- Then, this graph is shifted down by 4 units to create [tex]\( \sqrt[3]{x + 2} - 4 \)[/tex].
Resulting Transformation:
- Therefore, the graph of [tex]\( g(x) = \sqrt[3]{x + 2} - 4 \)[/tex] is the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] shifted left by 2 units and down by 4 units.
This transformation has been carefully described and can be visualized accordingly. There isn't a specific option given (A or B) in the problem, but it should be clear that:
"The graph of [tex]\( g(x) = \sqrt[3]{x + 2} - 4 \)[/tex] is the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] shifted left by 2 units and down by 4 units."
