Graph each function over a one-period interval.
y = -1 + csc x

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Step 1: Understand the function y = -1 + \(\csc\) x. The function \(\csc\) x is the cosecant function, which is the reciprocal of the sine function, \(\csc\) x = \(\frac{1}{\sin x}\).

Step 2: Identify the period of the function \(\csc\) x. Since \(\csc\) x is derived from \(\sin\) x, it has the same period as \(\sin\) x, which is 2\(\pi\).

Step 3: Determine the vertical shift. The function y = -1 + \(\csc\) x indicates a vertical shift of -1 unit. This means the entire graph of \(\csc\) x is shifted downward by 1 unit.

Step 4: Identify the vertical asymptotes. The function \(\csc\) x has vertical asymptotes where \(\sin\) x = 0, which occur at x = n\(\pi\), where n is an integer. These asymptotes will also apply to y = -1 + \(\csc\) x.

Step 5: Sketch the graph over one period. Start from x = 0 to x = 2\(\pi\), plot the vertical asymptotes, and sketch the transformed \(\csc\) x curve, considering the vertical shift of -1.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x). The cosecant function has a range of all real numbers except for values between -1 and 1, and it is undefined wherever sin(x) equals zero. Understanding its behavior is crucial for graphing functions that involve csc(x).

Vertical Shifts

Vertical shifts occur when a constant is added to or subtracted from a function. In the given function y = -1 + csc(x), the '-1' indicates a downward shift of the entire cosecant graph by one unit. This transformation affects the function's range and the position of its asymptotes, which are critical for accurately graphing the function.

Periodicity of Trigonometric Functions

Trigonometric functions, including the cosecant function, are periodic, meaning they repeat their values in regular intervals. The period of csc(x) is 2π, which means the function will complete one full cycle over this interval. Recognizing the periodic nature of the function is essential for graphing it accurately over a one-period interval.

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