the graph of $y = \\tan\\left\\frac{1}{4}\\left(x - \\frac{\\pi}{2}\\right)\\right+1$ is shown. what is the…

Explanation:

Step1: Recall period formula for tangent function

The general form of a tangent function is $y = A\tan(B(x - C))+D$, and its period is given by $T=\frac{\pi}{|B|}$. For the function $y=\tan\left[\frac{1}{4}(x - \frac{\pi}{2})\right]+1$, here $B = \frac{1}{4}$.

Step2: Calculate the period

$T=\frac{\pi}{\left|\frac{1}{4}\right|}=4\pi$.

Step3: Recall asymptote formula for tangent function

The asymptotes of the tangent function $y = A\tan(B(x - C))+D$ occur at $B(x - C)=\frac{\pi}{2}+n\pi$, where $n\in\mathbb{Z}$.

Step4: Solve for $x$ to find asymptotes

For $y=\tan\left[\frac{1}{4}(x - \frac{\pi}{2})\right]+1$, we set $\frac{1}{4}(x - \frac{\pi}{2})=\frac{\pi}{2}+n\pi$. First, multiply both sides by 4: $x-\frac{\pi}{2}=2\pi + 4n\pi$. Then, add $\frac{\pi}{2}$ to both sides: $x=\frac{\pi}{2}+2\pi+4n\pi=\frac{5\pi}{2}+4n\pi$, where $n\in\mathbb{Z}$.

Answer:

Period: $4\pi$ Asymptotes: $x = \frac{5\pi}{2}+4n\pi,n\in\mathbb{Z}$