In Exercises 17–24, graph two periods of the given cotangent function. y = 1/2 cot 2x

Verified step by step guidance

1. Recall the general form of the cotangent function: \(y = A \cot(Bx)\), where \(A\) is the amplitude factor (vertical stretch/compression) and \(B\) affects the period of the function.
2. Identify the period of the cotangent function using the formula for period: \(\text{Period} = \frac{\pi}{B}\). Here, \(B = 2\), so the period is \(\frac{\pi}{2}\).
3. Since the problem asks for two periods, calculate the total length along the x-axis to graph: \(2 \times \frac{\pi}{2} = \pi\).
4. Note the vertical stretch factor \(A = \frac{1}{2}\), which means the cotangent values will be scaled by \(\frac{1}{2}\) vertically. This affects the height of the graph but not the period or zeros.
5. To graph, plot key points within one period: cotangent has vertical asymptotes where \(\sin(2x) = 0\), i.e., at \(x = 0, \frac{\pi}{2}, \pi, \ldots\). Between these asymptotes, plot the cotangent curve scaled by \(\frac{1}{2}\), then repeat for the second period.

Key Concepts

Cotangent Function and Its Graph

The cotangent function, cot(x), is the reciprocal of the tangent function and is defined as cos(x)/sin(x). Its graph has vertical asymptotes where sin(x) = 0, and it repeats every π units. Understanding the shape and behavior of cotangent is essential for graphing transformations.

Period of a Trigonometric Function

The period of a function is the length of one complete cycle before it repeats. For cot(bx), the period is π divided by the absolute value of b. In this problem, with cot(2x), the period is π/2, which affects how the graph is drawn over the x-axis.

Amplitude and Vertical Scaling

Although cotangent functions do not have a maximum or minimum amplitude like sine or cosine, vertical scaling affects the steepness of the graph. The coefficient 1/2 in y = (1/2) cot(2x) compresses the graph vertically, making the slopes less steep compared to the standard cotangent curve.