In Exercises 17–24, graph two periods of the given cotangent function.y = 2 cot x

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1. Identify the basic form of the cotangent function, which is \( y = a \cdot \cot(bx - c) + d \). In this case, \( a = 2 \), \( b = 1 \), \( c = 0 \), and \( d = 0 \).
2. Determine the period of the function. The period of \( \cot(bx) \) is \( \frac{\pi}{b} \). Since \( b = 1 \), the period is \( \pi \).
3. Identify the vertical asymptotes of the function. For \( \cot(x) \), the vertical asymptotes occur at \( x = n\pi \), where \( n \) is an integer. Therefore, for \( y = 2 \cot x \), the asymptotes are at \( x = n\pi \).
4. Sketch the graph of one period of \( y = 2 \cot x \) between two consecutive vertical asymptotes, such as from \( x = 0 \) to \( x = \pi \). The graph will start from positive infinity at \( x = 0 \), cross the x-axis at \( x = \frac{\pi}{2} \), and approach negative infinity at \( x = \pi \).
5. Repeat the graph for a second period, from \( x = \pi \) to \( x = 2\pi \), following the same pattern as the first period.

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 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.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their values over a specified interval. For the cotangent function, key features include vertical asymptotes where the function is undefined (at multiples of π), and x-intercepts at integer multiples of π. Recognizing these features helps in accurately sketching the graph.

Amplitude and Vertical Stretch

In the function y = 2 cot(x), the coefficient 2 indicates a vertical stretch of the cotangent function. While the cotangent function itself does not have an amplitude like sine or cosine, this factor affects the steepness of the graph. Understanding how vertical transformations impact the graph is essential for accurate representation.