Graph each function over a one-period interval.

y = csc (x - π/4)

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Identify the function: The given function is \( y = \csc(x - \frac{\pi}{4}) \). The cosecant function, \( \csc(x) \), is the reciprocal of the sine function, \( \sin(x) \).

Determine the period: The period of the cosecant function is the same as the sine function, which is \( 2\pi \).

Identify the phase shift: The expression \( x - \frac{\pi}{4} \) indicates a phase shift to the right by \( \frac{\pi}{4} \).

Determine the vertical asymptotes: Since \( \csc(x) = \frac{1}{\sin(x)} \), the vertical asymptotes occur where \( \sin(x - \frac{\pi}{4}) = 0 \). Solve \( x - \frac{\pi}{4} = n\pi \) for \( x \), where \( n \) is an integer.

Graph the function: Plot the vertical asymptotes and sketch the cosecant curve, which will have branches approaching the asymptotes. The graph will repeat every \( 2\pi \) and will be shifted to the right by \( \frac{\pi}{4} \).

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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 is undefined wherever the sine function is zero, which occurs at integer multiples of π. Understanding the behavior of the sine function is crucial for graphing the cosecant function, as it directly influences its shape and asymptotes.

Phase Shift

Phase shift refers to the horizontal translation of a trigonometric function along the x-axis. In the function y = csc(x - π/4), the term (x - π/4) indicates a phase shift of π/4 units to the right. This shift affects the position of the graph, moving all features, including asymptotes and intercepts, accordingly. Recognizing how phase shifts alter the graph is essential for accurate representation.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their values over a specified interval, typically one period. For the cosecant function, it is important to identify key points, asymptotes, and the overall shape of the graph. The period of the cosecant function is 2π, and understanding how to find and represent these features is vital for creating an accurate graph over the given interval.

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