To determine the correct transformation of the graph [tex]\( f(x) = |x+2| - 5 \)[/tex], let's understand how the transformations affect the basic graph of [tex]\( g(x) = |x| \)[/tex].
1. Horizontal Shifts:
- If we have [tex]\( |x + c| \)[/tex], where [tex]\( c \)[/tex] is positive, it represents a shift to the left by [tex]\( c \)[/tex] units.
- If we have [tex]\( |x - c| \)[/tex], where [tex]\( c \)[/tex] is positive, it represents a shift to the right by [tex]\( c \)[/tex] units.
2. Vertical Shifts:
- If we add a constant [tex]\( d \)[/tex] to [tex]\( |x| \)[/tex] outside the absolute value, [tex]\( |x| + d \)[/tex], it represents a shift up by [tex]\( d \)[/tex] units.
- If we subtract a constant [tex]\( d \)[/tex] from [tex]\( |x| \)[/tex] outside the absolute value, [tex]\( |x| - d \)[/tex], it represents a shift down by [tex]\( d \)[/tex] units.
Given the function [tex]\( f(x) = |x+2| - 5 \)[/tex]:
- The term [tex]\( |x+2| \)[/tex] indicates a horizontal shift. Since [tex]\( +2 \)[/tex] is inside the absolute value, it means the graph of [tex]\( |x| \)[/tex] is shifted to the left by 2 units.
- The term [tex]\( -5 \)[/tex] outside the absolute value indicates a vertical shift. Since it is [tex]\( -5 \)[/tex], it means the graph of [tex]\( |x| \)[/tex] is shifted down by 5 units.
Therefore, the correct transformation of the graph is that the graph of [tex]\( f(x) = |x| \)[/tex] was shifted to the left 2 units and down 5 units.
Hence, the correct choice is:
- The graph of [tex]\( f(x)=|x| \)[/tex] was shifted to the left 2 units, down 5 units.
This corresponds to the first option.
