directions: determine period, vertical shift, and asymptotes for each of the following. determine if there…

Explanation:

Step1: Find the period

For the tangent function $y = A\tan(Bx - C)+D$, the period is given by $\frac{\pi}{|B|}$. Here $B = 1$, so the period $T=\frac{\pi}{|1|}=\pi$.

Step2: Check for reflection

The general form of a tangent - function is $y = A\tan(Bx - C)+D$. Since $A = 2>0$, there is no reflection about the $x$ - axis.

Step3: Determine the vertical shift

Comparing $y = 2\tan(x-\frac{\pi}{4})$ with $y = A\tan(Bx - C)+D$, we have $D = 0$, so there is no vertical shift.

Step4: Find the asymptotes

The asymptotes of the tangent function $y=\tan x$ occur at $x=-\frac{\pi}{2}+k\pi$ and $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$. For the function $y = 2\tan(x-\frac{\pi}{4})$, we set $x-\frac{\pi}{4}=-\frac{\pi}{2}+k\pi$ and $x-\frac{\pi}{4}=\frac{\pi}{2}+k\pi$. For $x-\frac{\pi}{4}=-\frac{\pi}{2}+k\pi$, we get $x=-\frac{\pi}{4}+k\pi$. For $k = 0$, one asymptote is $x =-\frac{\pi}{4}$. For $x-\frac{\pi}{4}=\frac{\pi}{2}+k\pi$, we get $x=\frac{3\pi}{4}+k\pi$. For $k = 0$, the other asymptote is $x=\frac{3\pi}{4}$.

Step5: Find the 5 key - points

The period is $\pi$, and the interval for one period is $(-\frac{\pi}{4},\frac{3\pi}{4})$.

• When $x = 0$: $y = 2\tan(0-\frac{\pi}{4})=2\times(-1)=-2$.
• When $x=\frac{\pi}{4}$: $y = 2\tan(\frac{\pi}{4}-\frac{\pi}{4})=2\tan(0)=0$.
• When $x=\frac{\pi}{2}$: $y = 2\tan(\frac{\pi}{2}-\frac{\pi}{4})=2\tan(\frac{\pi}{4})=2$.
• The endpoints of the period interval $x =-\frac{\pi}{4}$ and $x=\frac{3\pi}{4}$ make the function undefined.

Answer:

Period: $\pi$ Reflection: no Vertical Shift: 0 (no vertical shift) Asymptotes: $x =-\frac{\pi}{4}$ and $x=\frac{3\pi}{4}$ Key - points: $(0, - 2),(\frac{\pi}{4},0),(\frac{\pi}{2},2)$ (with endpoints $x =-\frac{\pi}{4}$ and $x=\frac{3\pi}{4}$ being undefined)