To determine which graph represents the function [tex]\( g(x) = \tan(x - 4) + 2 \)[/tex], we need to understand the transformations applied to the parent tangent function [tex]\( \tan(x) \)[/tex].
1. Horizontal Shift (Phase Shift):
- The function [tex]\( g(x) = \tan(x - 4) + 2 \)[/tex] involves a horizontal shift.
- The term [tex]\( (x - 4) \)[/tex] inside the tangent function indicates a phase shift to the right by 4 units.
- This means that every feature of the tangent graph (its asymptotes, intercepts, and trajectory between asymptotes) will be shifted 4 units to the right along the x-axis.
2. Vertical Shift:
- The addition of 2 outside the tangent function, [tex]\( +2 \)[/tex], indicates a vertical shift.
- This means that the entire graph of the transformed tangent function will be shifted up by 2 units.
To summarize:
- The graph of [tex]\( g(x) = \tan(x - 4) + 2 \)[/tex] will show a rightward shift of 4 units compared to [tex]\( \tan(x) \)[/tex].
- The graph will also be moved upward by 2 units.
Let's identify the graph:
- The characteristic shape of the tangent function will be maintained but shifted and elevated as described.
Unfortunately, without seeing graphs A and B, I cannot directly tell which is correct. However, you can use the given information to match the described transformations to the options.
Notice the following when determining the correct graph:
- Look for the vertical asymptotes of the tangent function. They should be shifted 4 units to the right.
- The whole curve, including where it crosses the y-axis, should be 2 units higher than it normally would be for [tex]\( \tan(x) \)[/tex].
By carefully comparing these transformations to the given graphs A and B, you will be able to select the graph that accurately represents [tex]\( g(x) = \tan(x - 4) + 2 \)[/tex].
