Problem 47 - Precalculus with Calculus Previews

Question

Find the amplitude, period, and phase shift of the given function. Sketch at least one cycle of the graph. $$ y=-4 \sin \left(\frac{\pi}{3} x-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

Amplitude: 4, Period: 6, Phase shift: 1 unit right.

Step 1: Identify the general form

The general form for a sine function is \( y = a \sin(bx - c) + d \). Our function is \( y = -4 \sin \left(\frac{\pi}{3} x - \frac{\pi}{3}\right) \). By comparing, \( a = -4 \), \( b = \frac{\pi}{3} \), \( c = \frac{\pi}{3} \), and \( d = 0 \).

Step 2: Find the amplitude

The amplitude is given by the absolute value of \( a \), which is \( |a| = |-4| = 4 \).

Step 3: Determine the period

The period of a sine function is calculated as \( \frac{2\pi}{b} \). For our function, \( b = \frac{\pi}{3} \), so the period is \( \frac{2\pi}{\left(\frac{\pi}{3}\right)} = 6 \).

Step 4: Calculate the phase shift

The phase shift is determined by \( \frac{c}{b} \). In our function, \( c = \frac{\pi}{3} \) and \( b = \frac{\pi}{3} \), so the phase shift is \( \frac{\frac{\pi}{3}}{\frac{\pi}{3}} = 1 \). Since \( c \) is positive, the function shifts \( 1 \) unit to the right.

Step 5: Sketch the graph

To sketch the graph, note the amplitude of 4 means the graph will oscillate between -4 and 4, the period of 6 means one complete cycle occurs over an interval of 6, and a phase shift of 1 unit to the right. Begin plotting from \( x = 1 \), marking critical points every 1.5 units until you complete one cycle, ending at \( x = 7 \).

Key Concepts

Understanding Amplitude

The amplitude of a trigonometric function refers to the height of its peaks from its central axis. It shows how far up and down the graph moves. In the case of the sine function \[y = -4 \sin \left(\frac{\pi}{3} x - \frac{\pi}{3}\right)\], amplitude is determined by the absolute value of \(a\), which is the coefficient in front of the sine function.

• Here, \(|a| = |-4| = 4\).
• This signifies that the graph peaks 4 units above and goes 4 units below its midline, which is 0 in this case.

Amplitude serves as a measure of the function's 'loudness' or intensity. It's essential for understanding how dynamic the graph of the function is, marking the extent of oscillation from its resting state.

Determining the Period

The period of a trig function like sine or cosine is the horizontal length required for one complete cycle of the graph. It's determined by the value of \(b\) in the function \(y = a \sin(bx - c)\). It's calculated using the formula:
\[\text{Period} = \frac{2\pi}{b}\]For the function \(y = -4 \sin \left(\frac{\pi}{3} x - \frac{\pi}{3}\right)\):

• \(b = \frac{\pi}{3}\)
• Therefore, \(\text{Period} = \frac{2\pi}{\left(\frac{\pi}{3}\right)} = 6\)

This means that every 6 units along the x-axis, the pattern of the graph repeats itself. Knowing the period helps you sketch graphs accurately by indicating where each cycle starts and ends. This periodic nature characterizes many natural phenomena, such as tides or sound waves.

Interpreting the Phase Shift

Phase shift refers to the horizontal movement of the graph from its usual position. It reveals how much the function is moved left or right on the graph.
For our function \[y = -4 \sin \left(\frac{\pi}{3} x - \frac{\pi}{3}\right)\], the phase shift is calculated as:

• \(\text{Phase Shift} = \frac{c}{b} = \frac{\frac{\pi}{3}}{\frac{\pi}{3}} = 1\).
• Since \(c\) is positive, the graph shifts to the right by 1 unit.

Understanding phase shift is crucial when positioning graphs relative to the y-axis. In phases like this, the graph begins its cycle not at the origin, but offset to the right, influencing when certain points of the wave will appear or disappear. This adjustment often models real-world situations where events start not at a baseline, but with an initial delay or displacement.