Problem 31 - Precalculus Essentials

Question

Graph two periods of the given cosecant or secant function. $$y=\frac{1}{2} \csc \frac{x}{2}$$

Step-by-Step Solution

Answer

To graph the function, first realize the original sine function has been modified - the amplitude has been halved and the period of function is twice as long. Then identify the period of the function which is \(4\pi\), and mark the vertical asymptotes at every \(x=2n\pi\). Now, graph the function such that it captures the 'U' shape of the cosecant function for two full periods.

Step 1: Understand the Basic Shape of a Cosecant Function

The cosecant function, \(csc(x)\), is the reciprocal of the sine function. It forms a 'U' shape with vertical asymptotes where the sine function is equal to zero.

Step 2: Identify the Modifications on the Original Function

The function \(y=\frac{1}{2} \csc \frac{x}{2}\) is a modified version of the basic cosecant function. The amplitude is halved because of the \(\frac{1}{2}\) factor, and the period is twice as long due to the \(\frac{x}{2}\) divisor.

Step 3: Identify the period of the function

For a basic trigonometric function like sine or cosine, the period is \(2\pi\). Since our function has a \(\frac{x}{2}\) factor, the period becomes \(2*(2\pi) = 4\pi\).

Step 4: Identify the Vertical Asymptotes

The cosecant function has vertical asymptotes where \(sin(x)=0\). In this case, that would be every \(x=2n\pi\), where \(n\) is an integer.

Step 5: Graph the Function

Plot the vertical asymptotes first. Then, plot key points for the cosecant function, ensuring you capture the 'U' shape. Also, ensure that you graph two full periods of the function.