Sure! Let's break down the transformation of the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] to the function [tex]\( g(x) = f(x+2) - 4 \)[/tex].
1. Parent Function: The parent function is given as [tex]\( f(x) = \sqrt[3]{x} \)[/tex]. This represents a cube root function.
2. Horizontal Shift: The function [tex]\( g(x) = f(x+2) - 4 \)[/tex] modifies the input of the parent function.
- [tex]\( f(x + 2) \)[/tex] indicates a horizontal shift. Specifically, [tex]\( x + 2 \)[/tex] means that every [tex]\( x \)[/tex] value is shifted to the left by 2 units. This is because adding inside the function moves the graph in the opposite direction of the sign.
3. Vertical Shift: The function [tex]\( g(x) = f(x+2) - 4 \)[/tex] also modifies the output of the parent function.
- The subtraction by 4 ([tex]\( - 4 \)[/tex]) outside the function indicates a vertical shift downward. This means that the entire graph is shifted downward by 4 units.
In summary:
- The transformation [tex]\( f(x + 2) \)[/tex] shifts the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] 2 units to the left.
- The transformation [tex]\( - 4 \)[/tex] shifts the graph downward by 4 units.
Thus, the graph of [tex]\( g(x) = \left( \sqrt[3]{x + 2} \right) - 4 \)[/tex] is a vertical shift of the cube root function [tex]\( x^{1/3} \)[/tex] down by 4 units and a horizontal shift to the left by 2 units.
Therefore, to select the correct graph [tex]\( g(x) \)[/tex] from the given options, look for the graph that has the cube root shape and is moved 2 units to the left and 4 units down.
