Problem 51 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The function \( y = |\cot(\frac{1}{2}x)| \) has a period of \( 2\pi \) and the graph starts from positive infinity, dips down to 0 at halfway point of the period and then goes up again to positive infinity. Repeat this over two periods.

Step 1: Understanding the Function

We are working with the function \( y = |\cot(\frac{1}{2}x)| \). This is the absolute value of the cotangent of \( \frac{1}{2}x \). The cotangent is the reciprocal function of the tangent and they share the same period but behave differently. The cotangent approaches infinity whenever its input approaches a multiple of pi and is undefined when its input is a multiple of pi. When the function is enclosed in an absolute value, it becomes positive everywhere.

Step 2: Determining the Period

The basic cotangent function, \( y = \cot(x) \), has a period of \( \pi \). However our function input has been halved, which by rule doubles the period of the function under normal circumstance. Hence, the period of the function \( y = |\cot(\frac{1}{2}x)| \) is \( 2\pi \).

Step 3: Plotting the graph

We will follow these steps to plot two periods of the graph: 1. Firstly, we draw vertical asymptotes (lines on which the function is undefined) at every multiple of \( 2\pi \). 2. We then plot points at halfway between these asymptotes. 3. In absence of any horizontal shifts, the cotangent function starts from infinity to negative infinity or vice versa, but in this case, due to the absolute value, the function will always remain positive. Therefore, it starts at positive infinity (from top), dips down to 0 at halfway point, and then goes up again to positive infinity (towards top). We repeat this for both periods.