To determine how to shift the graph of [tex]\( f(x) = |x| \)[/tex] to obtain the graph of [tex]\( g(x) = |x| - 9 \)[/tex], we need to analyze the transformation applied to [tex]\( f(x) \)[/tex].
1. Understand the basic graph of [tex]\( f(x) = |x| \)[/tex]:
The function [tex]\( f(x) = |x| \)[/tex] represents the absolute value function, which is V-shaped and symmetric about the y-axis. The vertex of the graph is at the origin (0,0), and it opens upwards.
2. Analyze the transformation to get [tex]\( g(x) = |x| - 9 \)[/tex]:
The function [tex]\( g(x) = |x| - 9 \)[/tex] can be interpreted as taking the original function [tex]\( f(x) = |x| \)[/tex] and subtracting 9 from it.
3. Interpret the effect of the subtraction:
Subtracting 9 from the function [tex]\( f(x) = |x| \)[/tex] translates the entire graph downward by 9 units. This is because, for every value of [tex]\( x \)[/tex], the output of [tex]\( g(x) \)[/tex] is 9 less than the output of [tex]\( f(x) \)[/tex]. Specifically, the new vertex of the graph shifts from (0,0) to (0,-9).
4. Determine the direction of the shift:
Since the subtraction leads to every point on the graph of [tex]\( f(x) \)[/tex] being moved 9 units down, the correct direction of the shift is down.
Hence, the graph of [tex]\( f(x) = |x| \)[/tex] must be shifted downwards to produce the graph of [tex]\( g(x) = |x| - 9 \)[/tex].
The correct answer is:
D. down
