To determine how the graph of [tex]\( f(x) = |x| \)[/tex] needs to be shifted to produce the graph of [tex]\( g(x) = |x| - 9 \)[/tex], let’s analyze the transformations applied to [tex]\( f(x) \)[/tex]:
1. The function [tex]\( f(x) = |x| \)[/tex] represents the absolute value function, which has its vertex at the origin (0,0) and opens upwards.
2. The function [tex]\( g(x) = |x| - 9 \)[/tex] can be understood as the absolute value function [tex]\( f(x) = |x| \)[/tex], but with a vertical shift.
3. To see how the graph shifts:
- Start with [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = |x|
\][/tex]
- Compare it with [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = |x| - 9
\][/tex]
- The term [tex]\(-9\)[/tex] indicates that each point on the graph of [tex]\( f(x) = |x| \)[/tex] is moved vertically downward by 9 units.
Therefore, to transform the graph of [tex]\( f(x) = |x| \)[/tex] to [tex]\( g(x) = |x| - 9 \)[/tex], you must shift the graph downward.
Thus, the correct direction is "down", which corresponds to choice:
C. down
