Problem 20 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The graph of y=2cot(2x) consists of curves shooting up towards positive infinity at x=0, approaching zero at x=\(\frac{\pi}{4}\), descending towards negative infinity at x=\(\frac{\pi}{2}\), and repeating this pattern for the second period at x=\(\pi\) and x=\(\frac{3\pi}{2}\).

Step 1: Identify the Period and Interval

The period of the function cot(x) is \(\pi\), but here we have a frequency of 2 which affects the period. Hence, the period of the function y = 2cot(2x) is \(\frac{\pi}{2}\). So, two periods of the function are covered in the interval \([0, 2\pi]\).

Step 2: Plot Key Points

Two periods of the function's cycle occurs every half-period, where the function changes from negative infinity to positive infinity, and back again. In one period of cot(x), the function goes from positive infinity at 0, crosses through 0 at \(\frac{\pi}{2}\), and then goes to negative infinity at \(\pi\). Given our period of \(\frac{\pi}{2}\), these events will happen at 0, \(\frac{\pi}{4}\), and \(\frac{\pi}{2}\) respectively, in just half of our period. The amplitude of the cotangent function is 2, which means it reaches a height of 2 (though the range is all real numbers for the cotangent function).

Step 3: Sketch the Graph

At x=0, the graph should start from positive infinity. It should approach zero without actually reaching it at x=\(\frac{\pi}{4}\). After this point, it should decrease towards negative infinity by x=\(\frac{\pi}{2}\). This pattern will repeat for the second half of the period, starting again from positive infinity at x=\(\pi\) and going to negative infinity by x=\(\frac{3\pi}{2}\). By connecting the points smoothly, the graph of two periods of the function y=2cot(2x) can be drawn.