Problem 52 - Precalculus Essentials

Question

Graph two periods of each function. $$y=\left|\tan \frac{1}{2} x\right|$$

Step-by-Step Solution

Answer

The graph of the function \(y=\left|\tan \frac{1}{2} x\right|\) will consist of a series of 'U' shaped curves, starting at \(x = n\pi\) where n is an integer, and meeting at each half-period of \(x = \pi / 2 + n\pi\). All y-values are positive due to the influence of the absolute function.

Step 1: Identify the Period of the function

The general tangent function, \(y = tan(x)\), has a period of \(\pi\). However, we have a constant \(1/2\) within the tangent function, which will make the period double. So, the period of the function \(y=\left|\tan \frac{1}{2} x\right|\) will be \(2\pi\). The function will repeat after an interval of \(2\pi\).

Step 2: Set the domain

Because the function is the absolute value of the tangent, it is defined for all real numbers except where the tangent function is undefined. The tangent function is undefined at \(x = \frac{\pi}{2} + n\pi\), where n is an integer. Therefore, the domain of the function is all real numbers except \(x = 2(\frac{\pi}{2} + n\pi)\) which simplifies to \(x = n\pi\).

Step 3: Determine the Pattern of the Tangent Function

An important characteristic of the tangent function is that it is positive in the first and third quadrants, and negative in the second and fourth quadrants. This will assist in establishing the pattern of \(y=\left|\tan \frac{1}{2} x\right|\), however, since we are considering the absolute value, all values will be positive.

Step 4: Consider Influence of the Absolute Function

Given that it's an absolute value function, all our y-values should be positive. So, even if the regular \(\tan(x)\) dips below the x-axis, due to the absolute value, those negative values will be reflected above the x-axis.

Step 5: Graph the Function

First, draw vertical asymptotic lines at every interval that equals the period. Meaning, the first asymptotic line will be at \(x = \pi\), then at \(x = 2\pi\), these are the points where the original tangent function goes to \(+\infty\) and \(-\infty\), they are unchanged in the absolute function because they are positive. Then, graph the function, making sure to reflect all negative y-values above the x-axis. The resulting graph will look like a series of 'U' shaped curves, starting at each vertical asymptote and meeting at each half-period.