Graph each function over a two-period interval. y = cot (3x + π/4)
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Identify the basic form of the cotangent function: . Here, and .
Determine the period of the function. The period of is . For , the period is .
Calculate the interval for two periods. Since one period is , two periods will be .
Determine the phase shift. The phase shift is given by . For this function, it is .
Graph the function over the interval to , marking key points such as vertical asymptotes and zeros of the cotangent function.
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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. It is defined as cot(x) = cos(x)/sin(x). The cotangent function has a period of π, meaning it repeats its values every π units along the x-axis. Understanding its behavior, including asymptotes and zeros, is crucial for graphing.
Phase shift refers to the horizontal shift of a periodic function due to a constant added to the variable. In the function y = cot(3x + π/4), the term π/4 indicates a leftward shift of the graph by π/12 units (since the coefficient of x is 3). Recognizing how phase shifts affect the graph's starting point is essential for accurate plotting.
Vertical stretch or compression occurs when a function is multiplied by a constant factor. In the function y = cot(3x + π/4), the coefficient 3 in front of x indicates a vertical compression of the cotangent function, making it oscillate more rapidly. This affects the frequency of the graph, which is important for determining the number of cycles within the specified interval.