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.

Key Concepts

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

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

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.