which of the following is the graph of $y = 0.25csc(x+pi)+1$?

Explanation:

Step1: Recall csc - function property

We know that $\csc(x+\pi)=-\csc(x)$. So, $y = 0.25\csc(x+\pi)+1=- 0.25\csc(x)+1$.

Step2: Analyze key - points of the basic $\csc$ - function

The basic cosecant function $y = \csc(x)$ has vertical asymptotes at $x = k\pi,k\in\mathbb{Z}$, and its range is $(-\infty,- 1]\cup[1,\infty)$. For $y=-0.25\csc(x)+1$, the vertical asymptotes remain at $x = k\pi,k\in\mathbb{Z}$. When $\csc(x)=-1$, $y=-0.25\times(-1)+1 = 1.25$. When $\csc(x)=1$, $y=-0.25\times1 + 1=0.75$.

Step3: Match with the graph

We look for a graph with vertical asymptotes at $x = k\pi$, and the local maxima and minima values around $y = 1.25$ and $y = 0.75$ respectively.

Answer:

(Without seeing all the options, we can't give a specific choice. But the above - described characteristics should be used to identify the correct graph among the given options.)