To determine the direction in which 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 start by understanding the transformations involved.
1. Begin with the given functions:
- [tex]\( f(x) = x \)[/tex]
- [tex]\( g(x) = f(x) - 4 \)[/tex]
2. Substitute [tex]\( f(x) \)[/tex] into the expression for [tex]\( g(x) \)[/tex]:
- Since [tex]\( f(x) = x \)[/tex], we substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = x - 4
\][/tex]
3. Interpret the transformation:
- The equation [tex]\( g(x) = x - 4 \)[/tex] indicates that for each value of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] outputs the value of [tex]\( f(x) \)[/tex] decreased by 4.
- On a graph, subtracting 4 from [tex]\( f(x) \)[/tex] corresponds to moving every point on the graph of [tex]\( f(x) \)[/tex] vertically downward by 4 units. This is because subtracting a constant from a function's output translates to a vertical shift downward.
4. Conclusion:
- To convert [tex]\( f(x) = x \)[/tex] into [tex]\( g(x) = x - 4 \)[/tex], the entire graph must be shifted downward by 4 units.
Thus, the graph of [tex]\( f(x) = x \)[/tex] must be shifted down to produce the graph of [tex]\( g(x) = f(x) - 4 \)[/tex].
Therefore, the correct answer is:
A. down
