Find the domain, range, amplitude, and period of y = -2 csc(-2x - π/4) - 1.

Steps to Solve

1. Identify the function and its characteristics The given function is ( y = -2 \csc(-2x - \frac{\pi}{4}) - 1 ). The cosecant function has certain properties that we need to analyze, including the transformations applied to it.

2. Determine the domain The cosecant function, ( \csc(x) ), is undefined wherever ( \sin(x) = 0 ). For ( \csc(-2x - \frac{\pi}{4}) ), we set ( -2x - \frac{\pi}{4} = n\pi ) where ( n \in \mathbb{Z} ):

   Rearranging gives:

   $$ -2x = n\pi + \frac{\pi}{4} $$ $$ x = -\frac{n\pi + \frac{\pi}{4}}{2} $$

   Thus, the domain is all ( x ) except these values.

3. Calculate the range The general range of the cosecant function is ( (-\infty, -1] ) and ( [1, \infty) ). The transformation ( -2 \csc ) reflects the function and scales it by 2, changing the range to ( (-\infty, -2] ) and ( [2, \infty) ). After translating down by 1, the final range becomes:

   $$ (-\infty, -3] \cup [1, \infty) $$

4. Find the amplitude The amplitude of a cosecant function is not defined in the traditional sense like sine or cosine because it can take on various values depending on the vertical stretch and reflection. Here, however, the coefficient ( -2 ) indicates a vertical stretch but reflects it over the x-axis. Thus, we consider the scaling but not technically "amplitude."

5. Determine the period The period of the cosecant function is determined by the coefficient of ( x ) in the argument of the cosecant function. For our function, the coefficient is ( -2 ). Therefore, the period is calculated as follows:

   $$ \text{Period} = \frac{2\pi}{| -2 |} = \frac{2\pi}{2} = \pi $$