To describe the transformation of the absolute value function given by:
[tex]\[ f(x) = |x + 1| - 3 \][/tex]
we need to understand how the function [tex]\( |x| \)[/tex] has been altered.
1. Horizontal Shift:
- The term [tex]\( x + 1 \)[/tex] inside the absolute value function indicates a horizontal shift. Generally, [tex]\( |x - h| \)[/tex] shifts the graph of [tex]\( |x| \)[/tex] horizontally by [tex]\( h \)[/tex] units. If [tex]\( h \)[/tex] is positive, it shifts to the right; if [tex]\( h \)[/tex] is negative, it shifts to the left. In our case, [tex]\( x + 1 \)[/tex] can be written as [tex]\( x - (-1) \)[/tex], which means the graph shifts to the left by 1 unit.
2. Vertical Shift:
- The constant term outside the absolute value function, [tex]\(-3\)[/tex], indicates a vertical shift. For the form [tex]\( |x| + k \)[/tex], if [tex]\( k \)[/tex] is positive, it shifts the graph up; if [tex]\( k \)[/tex] is negative, it shifts the graph down. Since we have [tex]\(-3\)[/tex], this means the graph shifts down by 3 units.
Combining these transformations, the function [tex]\( f(x) = |x + 1| - 3 \)[/tex] results in:
- A horizontal shift to the left by 1 unit.
- A vertical shift downward by 3 units.
So, the correct description of the transformation is:
Horizontal shift to the left 1 and vertical shift down 3.
