Graph each function over a two-period interval.
y = 1 - cot x

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Identify the period of the function. Since the function is \(y = 1 - \cot x\), recall that the cotangent function \(\cot x\) has a period of \(\pi\). Therefore, a two-period interval for \(\cot x\) is \(2\pi\).

Determine the interval over which to graph the function. For \(\cot x\), a natural choice is from \(0\) to \(2\pi\) to cover two full periods.

Analyze the behavior of \(\cot x\) within one period. \(\cot x = \frac{\cos x}{\sin x}\) has vertical asymptotes where \(\sin x = 0\), which occur at \(x = 0, \pi, 2\pi\). Between these points, \(\cot x\) decreases from \(+\infty\) to \(-\infty\).

Apply the transformation to the function: \(y = 1 - \cot x\). This means you take the cotangent graph, reflect it vertically (because of the minus sign), and then shift it upward by 1 unit.

Sketch the graph over the interval \([0, 2\pi]\) by plotting key points and asymptotes at \(x = 0, \pi, 2\pi\), noting the vertical asymptotes and the shifted values of the function. This will give you the graph of \(y = 1 - \cot x\) over two periods.

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주요 개념

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

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. Understanding its periodicity and behavior near asymptotes is essential for graphing.

Periodicity of Trigonometric Functions

The cotangent function has a fundamental period of π, meaning its values repeat every π units. Graphing over a two-period interval involves plotting the function from 0 to 2π or any interval of length 2π, capturing two full cycles of cot(x).

Vertical Shifts in Trigonometric Graphs

The function y = 1 - cot(x) involves a vertical shift of the cotangent graph by 1 unit upwards. This means every point on the cotangent curve is increased by 1, affecting the position of the graph but not its shape or period.

Graph each function over a two-period interval.y = 1 - cot x