Problem 19 - Precalculus Essentials

Question

Graph two periods of the given cotangent function. $$y=\frac{1}{2} \cot 2 x$$

Step-by-Step Solution

Answer

The graph of the function \(y=\frac{1}{2} \cot 2x\) is obtained by drawing two periods of the function with a period of \(\frac{\pi}{2}\), amplitude of 1, no vertical shift and a vertical compression due to the \(\frac{1}{2}\).

Step 1: Identify the period of the cotangent function

The period of the cotangent function can be determined from the coefficient on \(x\). Since the coefficient on \(x\) is 2, the period of the function is equal to \(\frac{\pi}{2}\).

Step 2: Identify the amplitude and vertical shift of the cotangent function

The amplitude of a cotangent function is always 1, and there is no vertical shift in this function.

Step 3: Identify the phase shift of the cotangent function

In the function \(y=\frac{1}{2} \cot 2x\), there is no phase shift, so the graph of the function starts at the regular cotangent starting point.

Step 4: Graph of the cotangent function

Using the period \(\frac{\pi}{2}\), begin by drawing two asymptotes at \(x=0\) and \(x=\frac{\pi}{2}\). This is one period of the function. Then draw the second period by extending the x-axis and drawing two more asymptotes at \(x=\pi\) and \(x=\frac{3\pi}{2}\). Now, knowing that the cotangent function starts at the top and drops down at the center to the bottom at the next asymptote, draw the curve from the asymptote at \(x=0\) to the center of the period, then drop it down to the next asymptote. Repeat this for the second period as well. Lastly since the cotangent function is multiplied by \(\frac{1}{2}\), it needs to be vertically compressed by that factor.