To determine the period of the function [tex]\( y = \tan \left[\frac{1}{4}\left(x - \frac{\pi}{2}\right)\right] + 1 \)[/tex], we need to analyze the transformation that has been applied to the standard tangent function.
1. Standard Tangent Function:
- The period of the standard tangent function [tex]\( y = \tan(x) \)[/tex] is [tex]\( \pi \)[/tex].
2. Transformation Analysis:
- The given function is [tex]\( y = \tan \left[\frac{1}{4}\left(x - \frac{\pi}{2}\right)\right] + 1 \)[/tex].
- The argument of the tangent function is [tex]\( \frac{1}{4}\left(x - \frac{\pi}{2}\right) \)[/tex].
3. Coefficient Adjustment:
- For a general tangent function of the form [tex]\( y = \tan(bx + c) \)[/tex], the period is given by [tex]\( \frac{\pi}{|b|} \)[/tex].
- Here, the function’s argument is [tex]\( \frac{1}{4}(x - \frac{\pi}{2}) \)[/tex], which can be expressed as [tex]\( \frac{1}{4}x - \frac{\pi}{8} \)[/tex].
- The coefficient [tex]\( b \)[/tex] of [tex]\( x \)[/tex] in [tex]\( \frac{1}{4}x - \frac{\pi}{8} \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
4. Period Calculation:
- To find the period, we divide [tex]\( \pi \)[/tex] by the absolute value of the coefficient of [tex]\( x \)[/tex]:
[tex]\[
\text{Period} = \frac{\pi}{\left|\frac{1}{4}\right|} = \frac{\pi}{\frac{1}{4}} = 4\pi
\][/tex]
Therefore, the period of the function [tex]\( y = \tan \left[\frac{1}{4}(x - \frac{\pi}{2})\right] + 1 \)[/tex] is [tex]\(\boxed{4\pi}\)[/tex].
