Given the function [tex]\( g(x) = \tan(x - 4) + 2 \)[/tex], let's analyze how it transforms the parent tangent function [tex]\( \tan(x) \)[/tex].
1. Horizontal Shift:
- The term [tex]\( (x - 4) \)[/tex] inside the tangent function indicates a horizontal shift.
- Specifically, [tex]\( x - 4 \)[/tex] means the function is shifted 4 units to the right.
2. Vertical Shift:
- The constant term [tex]\( +2 \)[/tex] outside the tangent function indicates a vertical shift.
- The function is shifted 2 units upward.
Therefore, the correct graph representing the function [tex]\( g(x) = \tan(x - 4) + 2 \)[/tex] will show the standard tangent graph shifted 4 units to the right and 2 units upward from the origin.
It’s important to look for these features in the graphs provided:
- Identify where the vertical asymptotes (which originally occur at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], for [tex]\( k \in \mathbb{Z} \)[/tex]) have shifted.
- Look for the point where the tangent function crosses its midline, which originally occurs at [tex]\( y = 0 \)[/tex] and has been shifted to [tex]\( y = 2 \)[/tex].
Thus, select the graph which demonstrates both a rightward shift by 4 units and an upward shift by 2 units for the tangent function.
