Problem 68 - Algebra and Trigonometry

Question

Graph one period of each function. $$y=\left|3 \cos \frac{2 x}{3}\right|$$

Step-by-Step Solution

Answer

The function y=\(|3 \cos(\frac{2x}{3})|\) has a period of \(3\pi\) and oscillates between 0 and 3, above the x-axis only.

Step 1: Understand the function

The given function is \(y=\left|3 \cos \frac{2 x}{3}\right|\). The outer function is the absolute value function which will make all y-values positive. The inner function is a cosine function, with a amplitude of 3 and a period of \(3\pi\). The amplitude is not going to affect the period of the function, but the absolute part will affect what the function looks like since all y-values will be positive.

Step 2: Determine the period of the function

For the standard cosine function, the period is \(2\pi\). However, this function has a horizontal stretch by a factor of \(\frac{3}{2}\), which will affect the period. The period of this function is given by the formula \(\frac{2\pi}{|\frac{2}{3}|}=3\pi\). This means the function completes a full cycle over an interval of length \(3\pi\).

Step 3: Graph the function

On your graph, mark off an interval of length \(3\pi\) on the x-axis. This will give you one full period of the function. Then, recall that the absolute value function makes all y-values positive. So, unlike the typical cosine function which oscillates above and below the x-axis, this function will oscillate above the x-axis only. Start at the peak (3), go down to the x-axis, go back up to the peak (3), go down to the x-axis, and finally go back up to the peak (3). This gives you one full period of the function.