In Exercises 29–44, graph two periods of the given cosecant or secant function. y = −2 csc πx

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Identify the given function: \(y = -2 \csc(\pi x)\). Recall that \(\csc(\theta) = \frac{1}{\sin(\theta)}\), so this function is related to the sine function.

Determine the period of the function. The general period of \(\csc(bx)\) is \(\frac{2\pi}{b}\). Here, \(b = \pi\), so the period is \(\frac{2\pi}{\pi} = 2\).

Since the problem asks for two periods, calculate the interval for \(x\) over which to graph: from \(0\) to \(4\) (because one period is 2, two periods is \(2 \times 2 = 4\)).

Find the key points where the sine function (and thus the cosecant function) is zero, because \(\csc(\theta)\) is undefined where \(\sin(\theta) = 0\). Solve \(\sin(\pi x) = 0\) for \(x\) in \([0,4]\), which occurs at integer values \(x = 0, 1, 2, 3, 4\).

Plot the vertical asymptotes at these points where the function is undefined. Then, plot the shape of \(y = -2 \csc(\pi x)\) between these asymptotes, remembering that the negative sign reflects the graph over the x-axis and the amplitude 2 stretches it vertically.

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주요 개념

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

Understanding the Cosecant Function

The cosecant function, csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is undefined where sin(x) = 0, leading to vertical asymptotes. Its graph consists of branches that approach these asymptotes and have minimum or maximum points where sine reaches ±1.

Effect of Transformations on Trigonometric Graphs

Multiplying the cosecant function by a constant, such as -2, affects its amplitude and orientation. The factor 2 stretches the graph vertically, making peaks and troughs twice as far from the x-axis, while the negative sign reflects the graph across the x-axis, inverting its shape.

Period of the Cosecant Function with Horizontal Scaling

The period of csc(x) is 2π, but when the input is scaled by a factor (like πx), the period changes. The period is calculated as 2π divided by the coefficient of x inside the function. For y = -2 csc(πx), the period is 2π/π = 2, so two periods span an interval of length 4 on the x-axis.