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

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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.

Identify the period of the cotangent function using the formula for period: \(\text{Period} = \frac{\pi}{B}\). Here, \(B = 2\), so the period is \(\frac{\pi}{2}\).

Since the problem asks for two periods, calculate the total length along the x-axis to graph: \(2 \times \frac{\pi}{2} = \pi\).

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.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.