7. graph the function tan(1/2(x + π/2))

Explanation:

Step1: Identify the general form of tangent - function

The general form of the tangent function is $y = A\tan(B(x - C))+D$. For the function $y=\tan\left(\frac{1}{2}(x+\frac{\pi}{2})\right)$, we have $A = 1$, $B=\frac{1}{2}$, $C=-\frac{\pi}{2}$, and $D = 0$.

Step2: Find the period

The period of the tangent function $y = A\tan(B(x - C))+D$ is given by $T=\frac{\pi}{|B|}$. Since $B=\frac{1}{2}$, the period $T=\frac{\pi}{\frac{1}{2}} = 2\pi$.

Step3: Find the vertical asymptotes

The vertical asymptotes of the tangent function $y=\tan u$ occur at $u=(n+\frac{1}{2})\pi$, where $n\in\mathbb{Z}$. For $u=\frac{1}{2}(x + \frac{\pi}{2})$, we set $\frac{1}{2}(x+\frac{\pi}{2})=(n+\frac{1}{2})\pi$. Solve for $x$: [ \begin{align*} \frac{1}{2}(x+\frac{\pi}{2})&=(n+\frac{1}{2})\pi\ x+\frac{\pi}{2}&=2(n+\frac{1}{2})\pi\ x+\frac{\pi}{2}&=(2n + \pi)\ x&=2n\pi+\frac{\pi}{2} \end{align*} ]

Step4: Find key - points

We can find some key - points by choosing values of $x$ between the vertical asymptotes. For example, when $x =-\frac{\pi}{2}$, $y=\tan\left(\frac{1}{2}(-\frac{\pi}{2}+\frac{\pi}{2})\right)=\tan(0) = 0$. We can then plot the vertical asymptotes $x = 2n\pi+\frac{\pi}{2},n\in\mathbb{Z}$, the key - points, and use the periodicity and the shape of the tangent function (it has a shape of a series of "S - shaped" curves between vertical asymptotes) to graph the function.

To graph the function:

1. Mark the vertical asymptotes $x=\frac{\pi}{2},x=\frac{\pi}{2}+ 2\pi,x=\frac{\pi}{2}-2\pi,\cdots$ on the $x$ - axis.
2. Plot the point $(-\frac{\pi}{2},0)$.
3. Sketch the graph of the tangent function between the vertical asymptotes with a period of $2\pi$, passing through the key - points and having the characteristic shape of the tangent function.

The graph of the function $y = \tan\left(\frac{1}{2}(x+\frac{\pi}{2})\right)$ is a tangent - type curve with period $2\pi$, vertical asymptotes at $x=2n\pi+\frac{\pi}{2},n\in\mathbb{Z}$, and passing through the point $(-\frac{\pi}{2},0)$.

Answer:

Graph the function with period $2\pi$, vertical asymptotes at $x = 2n\pi+\frac{\pi}{2},n\in\mathbb{Z}$, and passing through the point $(-\frac{\pi}{2},0)$ using the characteristic shape of the tangent function.