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

Verified step by step guidance

1. Recall that the cotangent function \( \cot x \) has a period of \( \pi \). Since the function is \( y = 2 \cot x \), the period remains \( \pi \) because the coefficient 2 only affects the amplitude, not the period.
2. Determine the interval for two periods. Since one period is \( \pi \), two periods will span an interval of \( 2\pi \). For example, you can graph from \( x = 0 \) to \( x = 2\pi \).
3. Identify the key points within one period where the cotangent function has vertical asymptotes and zeros. The cotangent function has vertical asymptotes where \( \sin x = 0 \), i.e., at \( x = 0, \pi, 2\pi, \ldots \), and zeros where \( \cos x = 0 \), i.e., at \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \).
4. Calculate the corresponding \( y \)-values for the function \( y = 2 \cot x \) at points between the asymptotes, using the formula \( y = 2 \cot x = 2 \frac{\cos x}{\sin x} \). This will help you plot the shape of the graph accurately.
5. Sketch the graph by drawing vertical asymptotes at \( x = 0, \pi, 2\pi \), plotting the calculated points, and drawing the cotangent curve between these asymptotes for two full periods.

Key Concepts

Cotangent Function and Its Properties

The cotangent function, cot(x), is the reciprocal of the tangent function and is defined as cos(x)/sin(x). It has vertical asymptotes where sin(x) = 0, i.e., at integer multiples of π, and its period is π. Understanding its shape and behavior is essential for graphing.

Amplitude and Vertical Stretch

In the function y = 2 cot x, the coefficient 2 vertically stretches the cotangent graph by a factor of 2. While cotangent does not have a maximum or minimum amplitude like sine or cosine, this factor affects the steepness of the curve between asymptotes.

Period of the Cotangent Function

The standard period of cot(x) is π, meaning the function repeats every π units. Graphing two periods involves plotting the function over an interval of length 2π, showing two complete cycles between vertical asymptotes.