what are the period and vertical shift of the cosecant function below? period: 2π; vertical shift: 1 unit up…

Explanation:

Step1: Recall period formula for cosecant

The general form of a cosecant function is $y = A\csc(Bx - C)+D$. The period of $y = \csc(Bx)$ is $\frac{2\pi}{|B|}$. For the standard - form cosecant function, without horizontal stretching or compression ($B = 1$), the period of $y=\csc(x)$ is $2\pi$. Looking at the graph, we can see that one complete cycle occurs from $0$ to $2\pi$, so the period is $2\pi$.

Step2: Determine vertical shift

The vertical shift of the function $y = A\csc(Bx - C)+D$ is given by $D$. The standard cosecant function $y=\csc(x)$ has a mid - line at $y = 0$. In the given graph, the mid - line of the cosecant function is at $y = 2$. So the vertical shift is 2 units up.

Answer:

period: $2\pi$; vertical shift: 2 units up