Graph each function over a one-period interval.
y = 1 - (1/2) csc (x - 3π/4)

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Identify the basic form of the cosecant function: \( y = a + b \cdot \csc(c(x - d)) \). In this case, \( a = 1 \), \( b = -\frac{1}{2} \), \( c = 1 \), and \( d = \frac{3\pi}{4} \).

Determine the period of the function. The period of \( \csc(x) \) is \( 2\pi \), so the period of \( \csc(c(x - d)) \) is \( \frac{2\pi}{c} = 2\pi \).

Identify the phase shift, which is determined by \( d \). The function is shifted to the right by \( \frac{3\pi}{4} \).

Determine the vertical shift and reflection. The function is shifted up by 1 unit and reflected vertically due to the negative sign in front of \( \frac{1}{2} \).

Graph the function by plotting key points and asymptotes over one period, considering the transformations: vertical shift, reflection, and phase shift.

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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 the interval (-1, 1) and is undefined where sin(x) = 0. Understanding its properties, including its vertical asymptotes and periodicity, is essential for graphing functions involving csc.

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Graphs of Secant and Cosecant Functions

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. In the given function, y = 1 - (1/2) csc(x - 3π/4), the term (x - 3π/4) indicates a horizontal shift to the right by 3π/4, while the coefficient -1/2 affects the vertical stretch and reflection. Recognizing these transformations is crucial for accurately graphing the function.

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Domain and Range of Function Transformations

Period of Trigonometric Functions

The period of a trigonometric function is the length of one complete cycle of the function. For the cosecant function, the standard period is 2π. However, transformations can alter the period; in this case, since there are no horizontal scaling factors, the period remains 2π. Understanding the period helps in determining the intervals over which to graph the function accurately.

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