To determine which graph represents the function [tex]\( g(x) = \tan(x - 4) + 2 \)[/tex], we need to carefully analyze the transformations applied to the parent tangent function [tex]\( f(x) = \tan(x) \)[/tex].
Here are the steps to understand the transformations:
1. Horizontal Shift:
- The function [tex]\( g(x) \)[/tex] includes [tex]\( x - 4 \)[/tex] inside the tangent function.
- This indicates a horizontal shift to the right by 4 units.
2. Vertical Shift:
- The function [tex]\( g(x) \)[/tex] has an additional [tex]\( +2 \)[/tex] outside the tangent function.
- This indicates a vertical shift up by 2 units.
Putting it all together, the graph of [tex]\( g(x) = \tan(x - 4) + 2 \)[/tex] would show the tangent curve shifted 4 units to the right and 2 units up from the standard [tex]\( \tan(x) \)[/tex] curve.
For your question, without seeing the graphs labeled as A and B, it's about recognizing these transformations:
- The graph should show the [tex]\( \tan(x) \)[/tex] curve moved to the right by 4 units.
- It should also show that the whole curve is shifted upwards by 2 units, so the typical midpoint (which is usually at [tex]\( y = 0 \)[/tex]) is instead at [tex]\( y = 2 \)[/tex].
Given these transformations, select the graph that matches these descriptions:
- The asymptotes of [tex]\( g(x) \)[/tex] will now appear at [tex]\( x = 4 + n\pi \)[/tex] for integers [tex]\( n \)[/tex] because of the right shift.
- The periodicity of the tangent function should remain the same with the period [tex]\( \pi \)[/tex], but adjusted for the horizontal shift.
- The central points (where the tangent curve crosses the vertical midline and the steep slope starts) will be shifted both horizontally and vertically.
Study the graphs A and B based on these points and select the correct one accordingly.
