Question

graph the function.
$y = - \frac { 1 } { 2 } \csc \left( x + \frac { \pi } { 2 } \
ight)$

Explanation:

Step 1: Recall the general form of the cosecant function

The general form of the cosecant function is \(y = A\csc(Bx - C)+D\). For the given function \(y =-\frac{1}{2}\csc(x+\frac{\pi}{2})\), we have \(A =-\frac{1}{2}\), \(B = 1\), \(C=-\frac{\pi}{2}\), \(D = 0\).

Step 2: Find the period

The period of the cosecant function \(y = A\csc(Bx - C)+D\) is \(T=\frac{2\pi}{|B|}\). Since \(B = 1\), the period \(T = 2\pi\).

Step 3: Find the phase - shift

The phase - shift is given by \(\frac{C}{B}\). Here, \(C=-\frac{\pi}{2}\) and \(B = 1\), so the phase - shift is \(-\frac{\pi}{2}\).

Step 4: Analyze the vertical stretch and reflection

The factor \(A=-\frac{1}{2}\) causes a vertical compression (since \(|A|=\frac{1}{2}<1\)) and a reflection about the \(x\) - axis.

Step 5: Graph the function

1. First, consider the parent function \(y=\csc(x)\). Its asymptotes are at \(x = n\pi\), \(n\in\mathbb{Z}\), and it has local maxima at \(x=\frac{\pi}{2}+ 2n\pi\) (\(y = 1\)) and local minima at \(x=\frac{3\pi}{2}+2n\pi\) (\(y=-1\)).
2. For the function \(y =-\frac{1}{2}\csc(x+\frac{\pi}{2})\), the asymptotes are at \(x+\frac{\pi}{2}=n\pi\), i.e., \(x=n\pi-\frac{\pi}{2}=(2n - 1)\frac{\pi}{2}\), \(n\in\mathbb{Z}\).
3. When \(x+\frac{\pi}{2}=\frac{\pi}{2}+2n\pi\) (i.e., \(x = 2n\pi\)), \(y=-\frac{1}{2}\). When \(x+\frac{\pi}{2}=\frac{3\pi}{2}+2n\pi\) (i.e., \(x=\pi + 2n\pi\)), \(y=\frac{1}{2}\).

Answer:

To graph \(y =-\frac{1}{2}\csc(x+\frac{\pi}{2})\):

• Mark the asymptotes at \(x=(2n - 1)\frac{\pi}{2}\), \(n\in\mathbb{Z}\).
• Plot the key points: when \(x = 2n\pi\), \(y=-\frac{1}{2}\); when \(x=\pi+2n\pi\), \(y=\frac{1}{2}\).
• Sketch the two - part curves (one part between \(x=(2n - 1)\frac{\pi}{2}\) and \(x=(2n + 1)\frac{\pi}{2}\)) that approach the asymptotes and pass through the key points, with the shape of the cosecant function (a "U" - shaped curve and an inverted "U" - shaped curve in each period), considering the vertical compression and reflection.