In Exercises 29–44, graph two periods of the given cosecant or secant function.
y = 2 sec(x + π)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Secant Function
The secant function, denoted as sec(x), is the reciprocal of the cosine function. It is defined as sec(x) = 1/cos(x). The secant function has vertical asymptotes where the cosine function is zero, leading to undefined values. Understanding the behavior of the secant function is crucial for graphing it accurately.
Periodic functions repeat their values in regular intervals, known as periods. For the secant function, the standard period is 2π, but transformations such as horizontal shifts and vertical stretches can alter this. In the given function y = 2 sec(x + π), the period remains 2π, but the graph is shifted left by π units and stretched vertically by a factor of 2.
Transformations involve shifting, stretching, or reflecting the graph of a function. In the function y = 2 sec(x + π), the '+ π' indicates a horizontal shift to the left, while the '2' indicates a vertical stretch. Understanding these transformations is essential for accurately graphing the function and predicting its behavior across its domain.