We start with the function [tex]\( f(x) = |x| \)[/tex], which is the absolute value function. To derive the function [tex]\( g(x) = |x + 9| - 5 \)[/tex], we need to apply a series of transformations to [tex]\( f(x) \)[/tex].
Let's analyze the transformations step by step:
1. Horizontal Shift:
The term [tex]\( |x + 9| \)[/tex] inside the absolute value indicates a horizontal shift. Since the expression is [tex]\( x + 9 \)[/tex], it means we shift the graph to the left by 9 units.
2. Vertical Shift:
The term [tex]\( -5 \)[/tex] outside the absolute value indicates a vertical shift. Subtracting 5 from the function shifts the graph downward by 5 units.
Thus, the function [tex]\( g(x) = |x + 9| - 5 \)[/tex] is obtained by shifting the function [tex]\( f(x) = |x| \)[/tex] down by 5 units and to the left by 9 units.
So to complete the description of the transformation for [tex]\( g \)[/tex]:
To get the function [tex]\( g \)[/tex], shift [tex]\( f \)[/tex] down by 5 units and to the left by 9 units.
