Verified Answer :
- The period of the function \(y = \csc \left(x - \frac{\pi}{4}\right) - 3\) is \(\boxed{2\pi}\).
- The asymptotes are located at \(\boxed{x = n\pi + \frac{\pi}{4}}\) for any integer \(n\).
- The range of the function is \(\boxed{y \leq -4}\) and \(\boxed{y \geq -2}\).
Explanation
- Analysis of the Function Properties
The function provided is \(y = \csc(x - \frac{\pi}{4}) - 3\). This is a transformed cosecant function of the form \(y = A \csc(B(x - C)) + D\), where \(A = 1\), \(B = 1\), \(C = \frac{\pi}{4}\), and \(D = -3\). We analyze the properties of this function by comparing it to the parent function \(f(x) = \csc(x)\).
- Calculating the Period
The period of a cosecant function \(y = A \csc(B(x - C)) + D\) is given by the formula \(\text{Period} = \frac{2\pi}{|B|}\). Since \(B = 1\), the period of the function is \(\frac{2\pi}{1} = 2\pi\).
- Locating the Asymptotes
Vertical asymptotes for the function \(y = \csc(\theta)\) occur whenever \(\sin(\theta) = 0\), which is at \(\theta = n\pi\) for any integer \(n\). In our case, the argument \(\theta = x - \frac{\pi}{4}\). Setting this equal to \(n\pi\) gives the equation \(x - \frac{\pi}{4} = n\pi\). Solving for \(x\), we find the asymptotes at \(x = n\pi + \frac{\pi}{4}\) for any integer \(n\).
- Determining the Range
The range of the parent cosecant function \(y = \csc(\theta)\) is \((-\infty, -1] \cup [1, \infty)\). The constant \(D = -3\) represents a vertical downward shift of 3 units. Applying this shift, the new range becomes \((-\infty, -1 - 3] \cup [1 - 3, \infty)\), which simplifies to \((-\infty, -4] \cup [-2, \infty)\). Thus, the range of the function is defined by \(y \leq -4\) or \(y \geq -2\).
- Final Answer Summary
In summary:
- The period of the function is \(2\pi\).
- The asymptotes occur at \(x = n\pi + \frac{\pi}{4}\) for any integer \(n\).
- The range is defined by \(y \leq -4\) and \(y \geq -2\).
Examples
In engineering, understanding the properties of oscillating functions like cosecant is essential when modeling signal interference or mechanical vibrations. For example, if a bridge's structural response to wind is modeled by a trigonometric function, finding the asymptotes allows engineers to identify points of potential structural instability where the amplitude becomes undefined. Calculating the period and range helps predict the frequency of these vibrations and the maximum expected displacement, ensuring safe infrastructure design.