In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 3 csc x

Verified step by step guidance

1. Recall that the cosecant function is the reciprocal of the sine function, so \(y = 3 \csc x\) can be written as \(y = \frac{3}{\sin x}\).
2. Identify the period of the basic \(\csc x\) function, which is \(2\pi\). Since there is no horizontal stretch or compression in \(y = 3 \csc x\), the period remains \(2\pi\).
3. To graph two periods, determine the interval over which to graph: from \(0\) to \(4\pi\) (since one period is \(2\pi\), two periods is \(4\pi\)).
4. Find the key points where \(\sin x = 0\) because \(\csc x\) is undefined there, causing vertical asymptotes. These occur at \(x = 0, \pi, 2\pi, 3\pi, 4\pi\) within the interval.
5. Plot the reciprocal values of \(\sin x\) scaled by 3, noting that the graph will have branches going to \(\pm \infty\) near the vertical asymptotes, and the minimum and maximum values of \(y = 3 \csc x\) correspond to \(y = \pm 3\) at points where \(\sin x = \pm 1\).

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 behavior helps in accurately sketching its graph.

Periodicity of Trigonometric Functions

The period of the basic cosecant function is 2π, meaning the function repeats its values every 2π units along the x-axis. Graphing two periods involves plotting the function over an interval of length 4π to capture two full cycles.

Amplitude and Vertical Stretch

The coefficient 3 in y = 3 csc x vertically stretches the graph by a factor of 3. This affects the distance of the graph's branches from the x-axis, increasing the minimum and maximum values of the function's range accordingly.