Let's analyze the transformation of the graph of the function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] with the given expression [tex]\( y = f(x - 7) + 3 \)[/tex].
1. Horizontal Transformation:
- The expression inside the function, [tex]\( (x - 7) \)[/tex], indicates a horizontal shift of the graph. For [tex]\( f(x - a) \)[/tex], the graph of [tex]\( f(x) \)[/tex] is shifted to the right by [tex]\( a \)[/tex] units if [tex]\( a \)[/tex] is positive, and to the left if [tex]\( a \)[/tex] is negative.
- In this case, [tex]\( x - 7 \)[/tex] means the graph is shifted 7 units to the right.
2. Vertical Transformation:
- The [tex]\( + 3 \)[/tex] outside the function indicates a vertical shift. For [tex]\( f(x) + b \)[/tex], the graph of [tex]\( f(x) \)[/tex] is shifted up by [tex]\( b \)[/tex] units if [tex]\( b \)[/tex] is positive and down if [tex]\( b \)[/tex] is negative.
- In this case, [tex]\( +3 \)[/tex] means the graph is shifted 3 units up.
Combining both transformations:
- The graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is shifted 7 units to the right.
- It is also shifted 3 units up.
Thus, the correct statement describing the transformation is:
D. It is the graph of [tex]\( f \)[/tex] translated 3 units up and 7 units to the right.
