In Exercises 29–44, graph two periods of the given cosecant or secant function.
y = −2 csc πx
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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 has vertical asymptotes where the sine function is zero, which occurs at integer multiples of π.
Graphing trigonometric functions involves understanding their periodic nature, amplitude, and transformations. The cosecant function has a period of 2π, and its graph consists of U-shaped curves that open upwards or downwards, depending on the sign of the function. Transformations such as vertical shifts, reflections, and stretches can alter the appearance of the graph.
The period of a trigonometric function is the length of one complete cycle of the graph. For the cosecant function, the period is determined by the coefficient of x in the argument of the sine function. In the given function y = -2 csc(πx), the period is 2, and the amplitude is represented by the coefficient of the cosecant, which indicates the vertical stretch and direction of the graph.