To compare the graph of [tex]\( g(x) = f(x) - 3 \)[/tex] with the graph of [tex]\( f(x) \)[/tex], we need to understand the transformation that is being applied to the function [tex]\( f(x) \)[/tex].
Let's analyze the function [tex]\( g(x) = f(x) - 3 \)[/tex]:
1. Vertical Shifts:
- When a constant is subtracted from a function, i.e., [tex]\( g(x) = f(x) - k \)[/tex], the effect on the graph of the function is a vertical shift.
- Specifically, [tex]\( g(x) = f(x) - 3 \)[/tex] means that each value of the function [tex]\( f(x) \)[/tex] is reduced by 3. This results in shifting the entire graph of [tex]\( f(x) \)[/tex] downward by 3 units.
2. Other Transformations (for completeness):
- Vertical Stretches/Compressions: Multiplying the function [tex]\( f(x) \)[/tex] by a constant factor [tex]\( a \)[/tex], i.e., [tex]\( g(x) = a \cdot f(x) \)[/tex], would stretch or compress the graph vertically. This is not the case here since we are subtracting a constant and not multiplying by one.
- Horizontal Shifts: Adding or subtracting a constant inside the function argument, i.e., [tex]\( g(x) = f(x - h) \)[/tex] or [tex]\( g(x) = f(x + h) \)[/tex], would shift the graph horizontally (to the left or the right). This is also not the case here since we are modifying the function value directly.
Given that [tex]\( g(x) = f(x) - 3 \)[/tex] causes a vertical shift:
- The graph of [tex]\( g(x) \)[/tex] is shifted downward by 3 units relative to the graph of [tex]\( f(x) \)[/tex].
Therefore, the correct statement is:
[tex]\[ \text{B. The graph of } g(x) \text{ is shifted 3 units down.} \][/tex]
