Problem 101 - Algebra and Trigonometry

Question

Use a graphing utility to graph two periods of the function. $$y=3 \sin (2 x+\pi)$$

Step-by-Step Solution

Answer

The graph of the function \(y=3\sin(2x+\pi)\) varies 3 units above and below the centerline, completes one full cycle from 0 to \(\pi\) and another from \(\pi\) to \(2\pi\) and is shifted left by \(\pi/2\) units.

Step 1: Identify the amplitude

The amplitude of the function \(y = a \sin(bx + c) + d\) is given by the absolute value of \(a\). For our function \(y = 3 \sin (2x + \pi)\), the amplitude is \(|3|\), which is 3. This means that the graph varies 3 units above and below the midline.

Step 2: Identify the period

The period of the function is given by \(2\pi/\abs{b}\). For our function, \(b = 2\), so the period of the function is \(2\pi/\abs{2} = \pi\). This means one complete cycle of the graph is completed in the interval of \(\pi\). Since we are asked for two periods, the graph has to cover the interval of \(2\pi\).

Step 3: Identify the phase shift

The phase shift of the function is given by \(-c/b\). For our function, \(c = \pi\), hence the phase shift is \(-\pi/2\). The graph will thus be shifted to the left by \(\pi/2\) units.

Step 4: Graph the function

Using the information gathered from steps 1 to 3, it's now possible to sketch the graph. It will rise and fall 3 units from the centreline. The function will complete one full cycle from 0 to \(\pi\) and the other complete cycle from \(\pi\) to \(2\pi\). The graph will also be shifted to the left by \(\pi/2\) units. With all this information in hand, the function is graphed using the graphing tool.