Problem 68 - Precalculus: A Concise Course

Question

Use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. $$ y=-4 \sin \left(\frac{2}{3} x-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

To graph the function \(y = -4\sin\left(\frac{2}{3}x - \frac{π}{3}\right)\) and visualize two full periods, identify the amplitude, period, and phase shift first. Amplitude is 4, period is \(3π\), and phase shift is \(π/2\). Use these values to accurately draw the function, and ensure that the graphing tool's window covers \(6π\) units (two full periods) in the x-direction.

Step 1: Identify the Amplitude, Period, and Phase Shift

This is a sine function in the form \(y = A\sin(Bx - C)\). For the given function \(y = -4\sin\left(\frac{2}{3}x - \frac{π}{3}\right)\), the amplitude is given by the absolute value of A, which is |-4| or 4. The period is given by the formula \(2π/B\), which is \(2π/(\frac{2}{3})\), or \(3π\). The phase shift, or horizontal shift, is given by \(C/B\), which is \(π/3 / (\frac{2}{3})\), yielding \(π/2\).

Step 2: Graph the Function

Use the graphing utility to graph the function. The amplitude is the distance from the maximum or minimum point to the midline, and here it's 4, which means the function will oscillate 4 units up and down from the midline. The period indicates how frequently the function repeats, and here it's \(3π\) which indicates that a full cycle is completed every \(3π\) units. The phase shift indicates the amount of horizontal shift, and as it's \(π/2\), the function will be shifted right \(π/2\) units.

Step 3: Choose an Appropriate Viewing Window

Finally, to visualize two full periods of the sine function, the chosen window needs to cover 2 periods in the x-direction. As the period of the given function is \(3π\), span from 0 to \(2 \times 3π\), which is \(6π\), to visualize two full periods

Key Concepts

Amplitude

The amplitude of a trigonometric function describes the height of its peaks and the depth of its troughs. In simpler terms, it tells us how far the highest or lowest point is from the center (or midline) of the wave. For the function \(y = -4 \sin \left(\frac{2}{3}x - \frac{\pi}{3}\right)\), the amplitude can be found by looking at the absolute value of the coefficient in front of the sine, which is \(-4\). Therefore, the amplitude is \(|−4| = 4\).

• This means the wave will reach as high as 4 units above and as low as 4 units below the midline.
• Amplitude reflects the vertical stretch or shrink of the function.

To imagine this, think of it as how loud a sound wave might "look" on an audio visualizer; higher amplitude means a louder sound.

Period

The period of a trigonometric function is the length it takes for the pattern to repeat itself. Essentially, the period tells us how long the wave is before it starts to duplicate its behavior. In the function \(y = -4 \sin \left(\frac{2}{3}x - \frac{\pi}{3}\right)\), the period is determined by the formula \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) within the sine function. Here, \(B = \frac{2}{3}\).

• Calculating the period involves \(\frac{2\pi}{\frac{2}{3}} = 3\pi\).
• This means the function completes one full cycle every \(3\pi\) units along the x-axis.

A visual analogy is to think of it as a spinning wheel; a full period is one complete rotation before the wheel's pattern repeats.

Phase Shift

The phase shift refers to the horizontal movement of a wave along the x-axis. It shows how much the wave has been shifted from its usual starting point. For the function \(y = -4 \sin \left(\frac{2}{3}x - \frac{\pi}{3}\right)\), the phase shift is calculated using the formula \(\frac{C}{B}\), where \(C\) is the horizontal shift term.

• In this scenario, \(C = \frac{\pi}{3}\) and \(B = \frac{2}{3}\).
• The calculation yields a phase shift of \(\frac{\frac{\pi}{3}}{\frac{2}{3}} = \frac{\pi}{2}\).

Consequently, the graph of the function moves to the right by \(\frac{\pi}{2}\) units.
Imagine phase shift as the starting line of a race; adjusting the start position backwards or forwards doesn't change the race distance, but alters where the race begins on the track.