In Exercises 13–16, the graph of a cotangent function is given. Select the equation for each graph from the following options:y = cot(x + π/2), y = cot(x + π), y = −cot x, y= −cot(x − π/2).

Verified step by step guidance

1. Identify the key features of the cotangent graph, such as the vertical asymptotes and x-intercepts.
2. Notice that the vertical asymptotes are at x = -2\pi/3 and x = \pi/3, which suggests a horizontal shift.
3. The x-intercepts are at x = -7\pi/6 and x = -\pi/6, indicating a phase shift.
4. Compare these features with the standard cotangent function y = \cot(x) and the given options.
5. Determine which equation matches the observed shifts and transformations in the graph.

Key Concepts

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 graph of the cotangent function has vertical asymptotes where sin(x) = 0, which occurs at integer multiples of π. Understanding the basic shape and periodicity of the cotangent function is essential for analyzing its transformations.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For the cotangent function, horizontal shifts can be represented by adding or subtracting a constant inside the function's argument, such as cot(x + π/2). Recognizing how these transformations affect the graph is crucial for identifying the correct equation from a given graph.

Vertical Asymptotes

Vertical asymptotes are lines that a graph approaches but never touches or crosses. For the cotangent function, vertical asymptotes occur at x = nπ, where n is an integer. In the provided graph, the locations of these asymptotes help determine the function's behavior and are key indicators for selecting the correct cotangent equation based on the graph's features.