precalculus quiz 4\n1. determine the features of the following function, then graph at least one period of…

Explanation:

Step1: Identify vertical stretch

The coefficient of the tangent function gives the vertical stretch. For $y = 2\tan(2(x-\frac{\pi}{4}))+1$, the vertical - stretch is 2.

Step2: Find the period

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

Step3: Determine phase shift

The phase - shift of the function $y = A\tan(Bx - C)+D$ is $\frac{C}{B}$. Here $C=\frac{\pi}{2}$ (since $Bx - C=2x-\frac{\pi}{2}$) and $B = 2$, so the phase shift is $\frac{\pi}{4}$ to the right.

Step4: Find vertical translation

The value of $D$ gives the vertical translation. For $y = 2\tan(2(x-\frac{\pi}{4}))+1$, the vertical translation is 1 unit up.

Step5: Find equations of asymptotes

The asymptotes of the tangent function $y=\tan x$ occur at $x = \frac{\pi}{2}+k\pi,k\in\mathbb{Z}$. For $y = 2\tan(2(x-\frac{\pi}{4}))+1$, we set $2(x-\frac{\pi}{4})=\frac{\pi}{2}+k\pi$. Solving for $x$: [ \begin{align*} 2(x-\frac{\pi}{4})&=\frac{\pi}{2}+k\pi\ 2x-\frac{\pi}{2}&=\frac{\pi}{2}+k\pi\ 2x&=\pi + k\pi\ x&=\frac{\pi}{2}(k + 1),k\in\mathbb{Z} \end{align*} ]

Answer:

Feature	Value
Vertical stretch	2
Period	$\frac{\pi}{2}$
Phase shift	$\frac{\pi}{4}$ to the right
Vertical translation	1 unit up
Equation of all asymptotes	$x=\frac{\pi}{2}(k + 1),k\in\mathbb{Z}$