which of the following is the graph of the function $y = 4\\csc(x)$?

To determine the graph of ( y = 4\csc(x) ), we analyze the properties of the cosecant function:

Step 1: Recall the definition of cosecant

The cosecant function is the reciprocal of the sine function: ( \csc(x) = \frac{1}{\sin(x)} ). Thus, ( y = 4\csc(x) = \frac{4}{\sin(x)} ).

Step 2: Identify the domain

The function is undefined where ( \sin(x) = 0 ), i.e., at ( x = n\pi ) for all integers ( n ) (vertical asymptotes at these points).

Step 3: Analyze the range

Since ( -1 \leq \sin(x) \leq 1 ) (and ( \sin(x) \neq 0 )), the range of ( \csc(x) ) is ( (-\infty, -1] \cup [1, \infty) ). For ( y = 4\csc(x) ), the range becomes ( (-\infty, -4] \cup [4, \infty) ) (the graph is stretched vertically by a factor of 4).

Step 4: Analyze the behavior in intervals between asymptotes

• When ( \sin(x) > 0 ) (e.g., in ( (0, \pi) )), ( \csc(x) > 0 ), so ( y = 4\csc(x) > 0 ). The function has a minimum value of ( 4 ) when ( \sin(x) = 1 ) (i.e., at ( x = \frac{\pi}{2} + 2n\pi )).
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