Graph each function over a two-period interval. y= -1 + (1/2) cot (2x - 3π)
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Identify the basic form of the cotangent function: . In this case, , , , and .
Determine the period of the function. The period of is . Here, , so the period is .
Calculate the phase shift using . Here, and , so the phase shift is . This means the graph is shifted to the right by .
Determine the vertical shift, which is given by . This means the entire graph is shifted down by 1 unit.
Graph the function over a two-period interval. Start by plotting key points of the basic function, apply the transformations (vertical stretch/compression, phase shift, and vertical shift), and repeat the pattern for two periods.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cotangent Function
The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). It is periodic with a period of π, meaning it repeats its values every π units. Understanding the behavior of the cotangent function is essential for graphing it accurately, especially when transformations are applied.
Transformations of functions involve shifting, stretching, or compressing the graph of a function. In the given function, y = -1 + (1/2) cot(2x - 3π), the '-1' indicates a vertical shift downward, while '(1/2)' represents a vertical compression. The '2x' inside the cotangent function indicates a horizontal compression, affecting the period of the function.
The period of a function is the length of one complete cycle of the function's graph. For the cotangent function, the standard period is π, but this can change with transformations. In this case, the '2x' in the cotangent function reduces the period to π/2, meaning the function will complete its cycle twice as fast, which is crucial for accurately graphing the function over the specified interval.