Problem 6 - Precalculus Essentials

Question

Graph two periods of the given tangent function. $$y=2 \tan \frac{x}{4}$$

Step-by-Step Solution

Answer

The graph begins at the point \(-2\pi, 0\), goes up to a vertical asymptote at \(x = 0\), then goes from \(0\) to \('2\pi, 0\). The same shape is repeated once to cover a total range of \(-2\pi\) to \(2\pi\). Thus, the graph of \(y = 2 \ tan(\frac{x}{4})\) over two periods is shown.

Step 1: Determine the Period

Firstly, determine the period of the function. The general tangent function has a period of \(\pi\). In this function, the period is affected by the \(\frac{1}{4}\) inside the tangent function. Therefore, the period would be \(\pi \div \frac{1}{4} = 4\pi\).

Step 2: Plot Key Points for One Period

Next, plot key points within one period of the function. The key points for the tangent function generally lie at -\(\frac{\pi}{2}\), 0, and \(\frac{\pi}{2}\). So for this function, the key points would be at quadruple these values, which are -\(2\pi\), 0, and \(2\pi\). For the value of the function at these points can be found by substituting these \(x\) values into \(y=2 \tan(\frac{x}{4})\). The results are as follow: at \(x=-2\pi\), \(y=0\); at \(x=0\), \(y=0\); and at \(x=2\pi\), \(y=0\).

Step 3: Draw the First Period

Once all the key points have been plotted, draw the curve of the first period of the function. The section from -\(2 \pi\) to 0 forms a part of the curve of the tangent function. Then, there is a vertical asymptote at \(x = 0\), after which the function curves up from 0 at \(x = 2\pi\).

Step 4: Draw the Second Period

Repeat the previous step to draw the second period of the function. The key points for the second period would be at \(2\pi\), \(4\pi\) and \(6\pi\). The function values for these points are the same as in one period.