To solve this problem, we need to understand how transformations affect the graph of a given function. The given function is [tex]\( f(x) = \sqrt[3]{x} \)[/tex], and we want to describe the graph of [tex]\( y = f(x-7) + 3 \)[/tex].
1. Horizontal Translation:
- A function [tex]\( f(x) \)[/tex] translated horizontally by [tex]\( c \)[/tex] units can be written as [tex]\( f(x - c) \)[/tex] for a translation to the right and [tex]\( f(x + c) \)[/tex] for a translation to the left.
- In our case, we have [tex]\( f(x - 7) \)[/tex], which indicates a horizontal translation 7 units to the right.
2. Vertical Translation:
- A function [tex]\( f(x) \)[/tex] translated vertically by [tex]\( d \)[/tex] units can be written as [tex]\( f(x) + d \)[/tex] for a translation upward and [tex]\( f(x) - d \)[/tex] for a translation downward.
- Here, we have [tex]\( f(x-7) + 3 \)[/tex], which indicates a vertical translation 3 units upward.
So, putting these together:
- [tex]\( f(x-7) \)[/tex]: translates the graph 7 units to the right.
- [tex]\( f(x-7) + 3 \)[/tex]: translates the resulting graph 3 units up.
Thus, the correct statement is:
D. It is the graph of [tex]\( f \)[/tex] translated 3 units up and 7 units to the right.
Therefore, D is the correct answer.
