To answer the question of how the graph of [tex]\( f(x) = x \)[/tex] must be shifted to produce the graph of [tex]\( g(x) = f(x) - 4 \)[/tex], let's carefully analyze the given functions.
First, let's recall some basic principles about graph transformations:
1. Vertical Shifts:
- Adding a constant [tex]\( k \)[/tex] to [tex]\( f(x) \)[/tex], such as in [tex]\( f(x) + k \)[/tex], shifts the graph of [tex]\( f(x) \)[/tex] upward by [tex]\( k \)[/tex] units.
- Subtracting a constant [tex]\( k \)[/tex] from [tex]\( f(x) \)[/tex], such as in [tex]\( f(x) - k \)[/tex], shifts the graph of [tex]\( f(x) \)[/tex] downward by [tex]\( k \)[/tex] units.
2. Horizontal Shifts:
- Adding a constant [tex]\( h \)[/tex] inside the function argument, such as in [tex]\( f(x + h) \)[/tex], shifts the graph of [tex]\( f(x) \)[/tex] to the left by [tex]\( h \)[/tex] units.
- Subtracting a constant [tex]\( h \)[/tex] inside the function argument, such as in [tex]\( f(x - h) \)[/tex], shifts the graph of [tex]\( f(x) \)[/tex] to the right by [tex]\( h \)[/tex] units.
Given the function [tex]\( f(x) = x \)[/tex]:
- The graph of [tex]\( f(x) \)[/tex] is a straight line passing through the origin with a slope of 1.
Now consider the function [tex]\( g(x) = f(x) - 4 \)[/tex]:
- Substituting [tex]\( f(x) \)[/tex] we get [tex]\( g(x) = x - 4 \)[/tex].
This function can be interpreted as the original function [tex]\( f(x) = x \)[/tex] with a vertical shift. Specifically, subtracting 4 from [tex]\( f(x) \)[/tex] means we shift the graph downward by 4 units.
Thus, the correct direction of the shift is downward.
The correct answer is:
B. down
