To determine which graph correctly represents the function [tex]\( g(x) = \tan(x-4) + 2 \)[/tex], we need to understand how this function transforms the parent tangent function [tex]\( f(x) = \tan(x) \)[/tex].
### Step-by-Step Explanation:
1. Horizontal Shift: The function [tex]\( g(x) = \tan(x-4) + 2 \)[/tex] includes a horizontal translation. The term [tex]\( (x-4) \)[/tex] indicates a shift to the right by 4 units. In other words, the entire graph of [tex]\( \tan(x) \)[/tex] is moved to the right by 4 units.
2. Vertical Shift: The function also has a vertical translation indicated by the [tex]\( +2 \)[/tex] outside of the tangent function. This means the entire graph is moved up by 2 units.
3. Asymptotes: The vertical asymptotes of the parent function [tex]\( \tan(x) \)[/tex] occur at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], where [tex]\( k \)[/tex] is any integer. When the function is shifted right by 4 units, the new asymptotes will be at [tex]\( x = \frac{\pi}{2} + 4 + k\pi \)[/tex].
4. Graph Characteristics:
- The characteristic shape of the tangent function remains the same.
- It has periodic vertical asymptotes and the function increases from [tex]\(-\infty\)[/tex] to [tex]\(+\infty\)[/tex] between the asymptotes.
- After the transformations, each original point of the tangent graph shifts right by 4 units and up by 2 units.
By considering these transformations, select the graph that:
- Shows a shift to the right by 4 units.
- Shows a shift up by 2 units.
- Has the correct shape and periodicity of the tangent function with altered asymptotes.
Choose the graph that meets these criteria succinctly. If needed, you can verify the appearance of the asymptotes and the center points of the periodic sections of the graph to ensure the correct transformations are accounted for.
