Question

graph the following function: $y = \frac{-1}{2}\csc(2x + \pi)$
step 1 of 2: identify the shape of the more basic function that has been shifted, reflected, stretched or compressed.

Explanation:

Step1: Recall basic trigonometric functions

The basic function related to $y = \frac{-1}{2}\csc(2x+\pi)$ is the cosecant function $y = \csc(x)$. The general form of a cosecant - type function is $y = A\csc(Bx - C)+D$. In our given function $y=\frac{-1}{2}\csc(2x+\pi)$, we can rewrite it as $y=\frac{-1}{2}\csc(2(x+\frac{\pi}{2}))+0$.

Step2: Identify the basic function shape

The shape of the basic cosecant function $y = \csc(x)$ has vertical asymptotes at $x = n\pi$, where $n\in\mathbb{Z}$, and it has U - shaped curves between consecutive asymptotes.

Answer:

The shape of the more basic function is that of the cosecant function, which has vertical asymptotes and U - shaped curves between consecutive asymptotes.