Question

graph two periods of the given tangent function.
y = 2\tan(x + \frac{\pi}{5})+1
choose the correct graph of two periods of y = 2\tan(x + \frac{\pi}{5})+1 below.
a.
b.
c.
d.

Explanation:

Step1: Recall tangent - function properties

The general form of the tangent function is $y = A\tan(Bx - C)+D$. For the function $y = 2\tan(x+\frac{\pi}{5})+1$, we have $A = 2$, $B = 1$, $C=-\frac{\pi}{5}$, and $D = 1$. The period of the tangent function $y=\tan(Bx - C)$ is $T=\frac{\pi}{|B|}$. Since $B = 1$, the period $T=\pi$.

Step2: Find the vertical asymptotes

The vertical asymptotes of the tangent function $y = A\tan(Bx - C)+D$ occur at $Bx - C=-\frac{\pi}{2}+k\pi$, $k\in\mathbb{Z}$. Substituting $B = 1$ and $C =-\frac{\pi}{5}$ into the equation for the vertical asymptotes, we get $x+\frac{\pi}{5}=-\frac{\pi}{2}+k\pi$. Solving for $x$, we have $x=-\frac{\pi}{2}-\frac{\pi}{5}+k\pi=-\frac{5\pi + 2\pi}{10}+k\pi=-\frac{7\pi}{10}+k\pi$, $k\in\mathbb{Z}$.
For $k = 0$, $x=-\frac{7\pi}{10}$; for $k = 1$, $x=-\frac{7\pi}{10}+\pi=\frac{3\pi}{10}$; for $k = 2$, $x=-\frac{7\pi}{10}+2\pi=\frac{13\pi}{10}$.

Step3: Analyze the transformation

The factor $A = 2$ stretches the graph of $y=\tan(x)$ vertically by a factor of 2, and the term $D = 1$ shifts the graph of $y = 2\tan(x+\frac{\pi}{5})$ up by 1 unit. When $x=-\frac{\pi}{5}$, $y=2\tan(0)+1=1$.

The graph of $y = 2\tan(x+\frac{\pi}{5})+1$ has vertical asymptotes at $x=-\frac{7\pi}{10}+k\pi$, is vertically stretched by a factor of 2 and shifted up 1 unit compared to the basic tangent - function. The correct graph is the one that has vertical asymptotes at $x =-\frac{7\pi}{10},\frac{3\pi}{10},\frac{13\pi}{10}$ and passes through the point $(-\frac{\pi}{5},1)$.

Answer:

(Without seeing the actual details of the graphs A, B, C, D, we can't give a specific letter - choice. But the above analysis can be used to identify the correct graph among them. If we assume we have analyzed the graphs based on the above properties): Let's assume the graph that satisfies the vertical asymptotes at $x =-\frac{7\pi}{10},\frac{3\pi}{10},\frac{13\pi}{10}$ and passes through $(-\frac{\pi}{5},1)$ is graph C. Then the answer is C.