Problem 68 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The graph of the function \(y = |3cos(\frac{2x}{3})|\) looks like waves on the positive side of the y-axis only (due to absolute function), has a peak at y = 3 (due to amplitude 3) and the period of the function is \(3\pi\) (due to multiplier 2/3).

Step 1: Identify the modifications

The modifications done to the normal cosine function are: \n1. The magnitude of the cosine is multiplied by 3, which will effect the amplitude of the function, making it from -3 to 3 instead of -1 to 1. \n2. Inside the cosine, the argument x is divided by \(3\), and then multiplied by \(2\). This will affect the period of the function. The normal period of the cosine function is \(2\pi\), this modification will make the period \(2\pi \times \frac{3}{2} = 3\pi\).

Step 2: Sketch the cosine function

Draw the cosine function with the identified modifications. The peak of the function will be at 3 (due to coefficient 3 of cosine) and the period should be \(3\pi\) (due to multiplier 2/3 inside the cosine). Plan the x-axis units according to the period \(3\pi\). For simplicity, divide the period into 4 equal parts: \(0,\frac{3\pi}{4}, \frac{3\pi}{2},\frac{9\pi}{4},3\pi\).

Step 3: Apply the absolute value

The function after the absolute sign will always be positive, which means the parts of the graph that fell under the x-axis will be reflected and appear above the x-axis giving the effect of absolute value.

Step 4: Simplify the graph

Now finally simplify the graph. Mark significant points which are the zero-crossings, maxima and minima on the graph. The resulting graph will oscillate between 0 and 3 and has a period \(3\pi\).

Step 5: Verify

Verify the graph using a graphing tool or make few calculations on significant values to compare with the graph.