Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=-4 \sin \left(\frac{2}{3} x-\frac{\pi}{3}\right)$$

Step-by-Step Solution

Answer

The amplitude of the function is 4, the period is \(3\pi\), and the phase shift is \(\frac{\pi}{3}\) to the right. When graphed, the function makes two full cycles between \(\frac{\pi}{3}\) and \(\frac{\pi}{3} + 2 \times 3\pi = 7\pi\).

Step 1: Identify the amplitude

The amplitude of a sine function is the absolute value of the coefficient of 'sin'. In this function it is the absolute value of '-4' i.e., \(|-4| = 4\) is the amplitude.

Step 2: Determine the frequency and period

The frequency of a sine function is just the number being multiplied by 'x' inside the sine. Here it is \(\frac{2}{3}\). The period of the function is the reciprocal of frequency. So we can calculate it as \(2 \pi \times \frac{3}{2} = 3\pi\). So, the period of the function is \(3\pi\).

Step 3: Determine the phase shift

The phase shift of the function is given by the constant term in the argument of sin, i.e., \(-\frac{\pi}{3}\). This indicates a shift of \(\frac{\pi}{3}\) units to the right.

Step 4: Plot the graph

Use a graphing tool to plot the sine function \(y = -4 \sin \left(\frac{2}{3} x-\frac{\pi}{3}\right)\) for two full periods. To do this, begin at the phase-shifted starting point and plot until you've completed two full periods.

Key Concepts

Amplitude and Period

When discussing trigonometric functions like sine and cosine, two essential characteristics are the amplitude and period. These describe how the graph of the function behaves over a cycle.

**Amplitude** refers to the height of the wave from the centerline to the peak. In mathematical terms, it's the absolute value of the coefficient in front of the sine function. For the function given in the exercise, \(y = -4 \sin \left(\frac{2}{3}x - \frac{\pi}{3}\right)\), the amplitude is \(|-4| = 4\). This means the waveform will reach 4 units above and below the central axis.

**Period** is the horizontal length of one complete cycle of the waveform. It's calculated as the reciprocal of the frequency of the sine function. The frequency is found as the coefficient of \(x\) inside the sine function, which in our case is \(\frac{2}{3}\). The formula to find the period \(T\) is \(T = \frac{2\pi}{|b|}\), where \(b\) is the frequency. For this function, the period is \(3\pi\), suggesting that a complete wave cycle occurs over a horizontal span of \(3\pi\) units.

Phase Shift

The phase shift indicates a horizontal movement of the graph along the \(x\)-axis. It shows us how much the function is shifted to the left or right from the typical starting point of zero.

In the function \(y = -4 \sin \left(\frac{2}{3}x - \frac{\pi}{3}\right)\), the phase shift is determined by examining the term \(-\frac{\pi}{3}\) inside the argument of the sine function. We rewrite it as \(\frac{2}{3}x - \frac{\pi}{3}\). The formula to find phase shift is \(\frac{c}{b}\), where \(c\) is the phase shift constant (\(-\frac{\pi}{3}\)) and \(b\) is the frequency (\(\frac{2}{3}\)).

So, the phase shift is \(- \left(\frac{\pi}{3}\right) / \left(\frac{2}{3}\right) = \frac{\pi}{3}\) to the right. This means the entire wave shifts \(\frac{\pi}{3}\) units horizontally to the right. Understanding phase shifts helps to accurately graph and analyze wave graphs in real-life applications.

Graphing Utility

A graphing utility is a tool, often a calculator or software, that helps visualize mathematical functions easily. These tools allow us to input equations and then generate the corresponding graph.

When plotting trigonometric functions like \(y = -4 \sin \left(\frac{2}{3}x - \frac{\pi}{3}\right)\) using a graphing utility, you can visualize how amplitude, period, and phase shift affect the graph:

• The amplitude will adjust the height of the graph above and below the axis (up to 4 units in this case).


• The period will influence the length of the graph required to complete a full cycle (3\(\pi\) units).


• A phase shift will move the graph horizontally across the \(x\)-axis (\(\frac{\pi}{3}\) units to the right).

Using a graphing utility is especially helpful as it allows you to graph two full periods with ease, ensuring you understand the behavior of the function over multiple cycles.