Final answer:
The function y = 4 csc(2x) has an amplitude of 4, a domain excluding points where 2x is an integer multiple of π, and its range includes all real numbers except those between -4 and 4.
Step-by-step explanation:
The student's question involves graphing the function y = 4 csc(2x) and determining its characteristics. To graph the cosecant function, which is the reciprocal of the sine function, one needs to remember that the cosecant graph will exist wherever the sine graph has non-zero values, as cosecant is undefined for sine values of zero.
The amplitude of a cosecant function is the absolute value of the coefficient in front of the csc term, which in this case is 4. The function's amplitude corresponds with the sine function's amplitude; namely, it is the distance from the midline to the maximum or minimum values of the wave.
However, since cosecant has vertical asymptotes where the sine function equals zero, these we treat as breaks in the graph. The domain of y = 4 csc(2x) excludes the points where 2x equals integer multiples of π since the sine function is zero at these points, and the cosecant function is undefined. The range of the function includes all real numbers except those between -4 and 4 because the absolute value of cosecant is always either greater than or equal to 1, or it is less than or equal to -1.