91Ó°ÊÓ

The period of \(\csc(x)\) is \(2\pi\). In the function \(y = 3\csc(4x)\), the coefficient of x inside the function is 4, which will shrink the graph along the x-axis (increases frequency). So, the period of the function is \(\frac{2\pi}{|4|} = \frac{\pi}{2}\).

The amplitude is not applicable to the cosecant function. So, there is no amplitude for this function.

In the function \(y = 3\csc(4x)\), the coefficient of the \(\csc(4x)\) is 3 which will stretch the graph along the y-axis. This means that every point (x, y) on the graph of \(\csc(x)\) will be transformed to the point (x, 3y) on the graph of the given function.

To sketch the graph of the function, first draw the graph of the basic function \(\csc(x)\) in the interval \([0, 2\pi]\). Then apply the transformations. Shrink the graph by a factor of 4 along the x-axis to get the period of \(\frac{\pi}{2}\) and stretch it by a factor of 3 along the y-axis. Since the period is \(\frac{\pi}{2}\), and the problem requires two periods, plot the graph in the interval \([0, \pi]\). Duplicate this in the next interval \([\pi, 2\pi]\) to get full two periods. The graph crosses the x-axis at multiples of \(\frac{\pi}{2}\). Also, sketch the vertical asymptotes at these points, the y-value increases as you move away from the vertical asymptote and decreases as you move towards the vertical asymptote.