Graph two periods of the given cosecant or secant function.


y = 2 csc x

Verified step by step guidance

1. Recall that the cosecant function is the reciprocal of the sine function, so \(\csc x = \frac{1}{\sin x}\). This means the graph of \(y = 2 \csc x\) is related to the graph of \(y = \sin x\), but with vertical stretches and undefined points where \(\sin x = 0\).
2. Identify the key points of the sine function over two periods. Since the period of \(\sin x\) is \(2\pi\), two periods span from \(0\) to \(4\pi\). Mark points where \(\sin x\) is 0, 1, or -1 within this interval, because these will influence the shape and asymptotes of \(y = 2 \csc x\).
3. Determine the vertical asymptotes of \(y = 2 \csc x\). These occur where \(\sin x = 0\), i.e., at \(x = 0, \pi, 2\pi, 3\pi, 4\pi\). Draw vertical dashed lines at these points to indicate the function is undefined there.
4. Plot the points of \(y = 2 \csc x\) by taking the reciprocal of the sine values and multiplying by 2. For example, where \(\sin x = 1\), \(y = 2 \times \frac{1}{1} = 2\), and where \(\sin x = -1\), \(y = 2 \times \frac{1}{-1} = -2\). Between the asymptotes, the graph will form branches that go to infinity near the asymptotes.
5. Sketch the two periods of \(y = 2 \csc x\) by connecting the plotted points with smooth curves that approach the vertical asymptotes, ensuring the graph reflects the reciprocal nature and vertical stretch of the function.

Key Concepts

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 at these points. Recognizing its periodicity and undefined points is essential for accurate graphing.

Amplitude and Vertical Stretch

In the function y = 2 csc x, the coefficient 2 acts as a vertical stretch, scaling the graph's values by a factor of 2. This affects the distance of the graph's peaks and troughs from the x-axis, making the graph taller compared to the basic csc(x) function.

Periodicity and Graphing Multiple Periods

The cosecant function has a period of 2π, meaning its pattern repeats every 2π units along the x-axis. Graphing two periods involves plotting the function from 0 to 4π (or an equivalent interval), including all asymptotes and key points to capture the full behavior twice.