the graph of y = csc(x - π/4) - 3 is shown. what is the period of the function? where are the asymptotes of…

Explanation:

Step1: Recall csc function period formula

The general form of a cosecant function is $y = A\csc(Bx - C)+D$, and its period is $T=\frac{2\pi}{|B|}$. For the function $y=\csc(x - \frac{\pi}{4})-3$, $B = 1$.

Step2: Calculate the period

$T=\frac{2\pi}{|1|}=2\pi$.

Step3: Find the asymptotes of csc function

The cosecant function $y = \csc(x)$ has asymptotes at $x = n\pi$, where $n\in\mathbb{Z}$. For the function $y=\csc(x - \frac{\pi}{4})-3$, we set $x-\frac{\pi}{4}=n\pi$. Solving for $x$ gives $x=n\pi+\frac{\pi}{4},n\in\mathbb{Z}$.

Step4: Determine the range of csc function

The range of the basic cosecant function $y = \csc(x)$ is $y\leq - 1$ or $y\geq1$. For the function $y=\csc(x - \frac{\pi}{4})-3$, we shift the basic csc - function down by 3 units. So the range is $y\leq-4$ or $y\geq - 2$.

Answer:

Period: $2\pi$ Asymptotes: $x = n\pi+\frac{\pi}{4},n\in\mathbb{Z}$ Range: $y\leq - 4$ $y\geq - 2$