To determine the transformation from [tex]\( f(x) = x^3 \)[/tex] to [tex]\( g(x) = (x+1)^3 \)[/tex], let's understand how the transformations affect the function.
1. Start with the given functions:
- [tex]\( f(x) = x^3 \)[/tex]
- [tex]\( g(x) = (x+1)^3 \)[/tex]
2. To interpret the transformation, recall that:
- A horizontal transformation involves changing the input [tex]\( x \)[/tex].
- [tex]\( g(x) = (x + c)^3 \)[/tex] indicates a horizontal shift of [tex]\( f(x) = x^3 \)[/tex].
- If [tex]\( c \)[/tex] is positive, the graph shifts [tex]\( |c| \)[/tex] units to the left.
- If [tex]\( c \)[/tex] is negative, the graph shifts [tex]\( |c| \)[/tex] units to the right.
3. Identify the shift in the function [tex]\( g(x) \)[/tex]:
- Here, [tex]\( g(x) \)[/tex] has the form [tex]\( (x + 1)^3 \)[/tex].
- The [tex]\( +1 \)[/tex] inside the function means we shift [tex]\( x \)[/tex] by -1 in the horizontal direction.
- This corresponds to moving the graph of [tex]\( f(x) \)[/tex] to the left by 1 unit.
Thus, the correct transformation is a horizontal shift of [tex]\( f(x) \)[/tex] one unit to the left.
The correct answer is:
B. a horizontal transformation of [tex]\( f(x) \)[/tex] one unit to the left.
