Question

1. graph two cycles of ( y = -cot(pi x - 2pi) - 1 ). label the scale on the ( x ) - and ( y ) - axes. plot the midpoint of each cycle, and label these points with ordered pairs. graph the asymptotes, and label these asymptotes with their equations. to the right, identify the period, and give the magnitude and direction of the horizontal shift and vertical shift. state the domain in set builder notation and the range in interval notation.

period
phase shift
vertical shift
domain
range

Explanation:

Step1: Find the period

The general form of the cotangent function is \(y = A\cot(Bx - C)+D\). For the function \(y=-\cot(\pi x - 2\pi)-1\), where \(B = \pi\). The period of the cotangent function \(y = A\cot(Bx - C)+D\) is given by \(T=\frac{\pi}{|B|}\).
Substituting \(B = \pi\) into the formula, we get \(T=\frac{\pi}{\pi}=1\).

Step2: Find the phase - shift

The phase - shift formula is \(h=\frac{C}{B}\). Here, \(C = 2\pi\) and \(B=\pi\), so \(h=\frac{2\pi}{\pi}=2\) (2 units to the right).

Step3: Find the vertical - shift

For the function \(y = A\cot(Bx - C)+D\), comparing with \(y=-\cot(\pi x - 2\pi)-1\), we have \(D=-1\) (1 unit down).

Step4: Find the domain

The cotangent function \(y = \cot(x)\) has a domain \(x
eq n\pi,n\in\mathbb{Z}\). For \(y =-\cot(\pi x - 2\pi)-1\), we set \(\pi x-2\pi
eq n\pi\).
Solve for \(x\):

$$
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$$

In set - builder notation, the domain is \(\{x|x
eq n + 2,n\in\mathbb{Z}\}\).

Step5: Find the range

The range of the cotangent function \(y=\cot(x)\) is \((-\infty,\infty)\). For \(y =-\cot(\pi x - 2\pi)-1\), since \(A=-1\) and \(D =-1\), the range is \((-\infty,\infty)\) (the vertical stretch and reflection do not change the range of the cotangent - type function).

Answer:

• Period: \(1\)
• Phase shift: \(2\) units to the right
• Vertical shift: \(1\) unit down
• Domain: \(\{x|x

eq n + 2,n\in\mathbb{Z}\}\)

• Range: \((-\infty,\infty)\)