Consider the following function from Example 5. Work these exercises in order. y = -2 - cot (x - π/4) Based on the answer in Exercise 58 and the fact that the cotangent function has period π, give the general form of the equations of the asymptotes of the graph of y = -2 - cot (x - π/4). Let n represent any integer.
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Identify the standard form of the cotangent function, which is \( y = a \cdot \cot(bx - c) + d \).
Recognize that the given function is \( y = -2 - \cot(x - \frac{\pi}{4}) \), which can be rewritten as \( y = -\cot(x - \frac{\pi}{4}) - 2 \).
Note that the cotangent function has vertical asymptotes where its argument is an integer multiple of \( \pi \), i.e., \( bx - c = n\pi \).
Substitute the values from the given function: \( x - \frac{\pi}{4} = n\pi \).
Solve for \( x \) to find the general form of the asymptotes: \( x = n\pi + \frac{\pi}{4} \).
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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 that cot(x + π) = cot(x) for any x. Understanding the behavior of the cotangent function is essential for analyzing its graph, particularly in identifying asymptotes and transformations.
Asymptotes are lines that a graph approaches but never touches. For the cotangent function, vertical asymptotes occur where the function is undefined, specifically at points where sin(x) = 0. In the case of y = -2 - cot(x - π/4), the asymptotes can be determined by finding the values of x that make cot(x - π/4) undefined, which occurs at x = π/4 + nπ, where n is any integer.
Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. In the given function y = -2 - cot(x - π/4), the term (x - π/4) indicates a horizontal shift to the right by π/4, while the '-2' indicates a vertical shift downward by 2 units. Understanding these transformations is crucial for accurately sketching the graph and identifying features such as asymptotes.