Graph each function over a two-period interval. y = 1 - cot x
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Step 1: Understand the function. The function given is . The cotangent function, , is the reciprocal of the tangent function, , and is defined as .
Step 2: Determine the period of . The cotangent function has a period of , meaning it repeats every units. Since we need to graph over a two-period interval, we will consider the interval .
Step 3: Identify key points and asymptotes. The cotangent function has vertical asymptotes where , which occur at . These are the points where the function is undefined.
Step 4: Transform the function. The given function is . This transformation involves a vertical shift of the cotangent graph upwards by 1 unit.
Step 5: Sketch the graph. Plot the key points and asymptotes on the interval . Draw the transformed cotangent curve, ensuring it approaches the asymptotes and reflects the vertical shift.
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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 radians. Understanding its behavior, including its asymptotes and zeros, is crucial for graphing functions that involve cotangent.
Transformations of functions involve shifting, stretching, or reflecting the graph of a function. In the case of y = 1 - cot(x), the graph of cot(x) is shifted vertically down by 1 unit. Recognizing how these transformations affect the original function's graph is essential for accurately plotting the new function.
Graphing a function over a specified interval, such as a two-period interval, requires understanding the function's periodicity and behavior within that range. For y = 1 - cot(x), since cot(x) has a period of π, a two-period interval would span from 0 to 2π. This involves plotting key points, identifying asymptotes, and ensuring the graph reflects the function's characteristics over the entire interval.