Problem 32 - Precalculus: Functions and Graphs

Question

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=4 \sin \left(\frac{1}{3} x-\frac{\pi}{3}\right)\)

Step-by-Step Solution

Answer

Amplitude: 4, Period: \(6\pi\), Phase Shift: \(\pi\) to the right.

Step 1: Identify the Amplitude

In the function \(y = A \sin(Bx - C)\), the amplitude is given by \(|A|\). For the function \(y = 4 \sin \left(\frac{1}{3}x - \frac{\pi}{3}\right)\), the amplitude is \(|4| = 4\).

Step 2: Determine the Period

The period of a sine function is given by \(\frac{2\pi}{|B|}\). Here, \(B = \frac{1}{3}\), so the period is \(\frac{2\pi}{\frac{1}{3}} = 6\pi\).

Step 3: Calculate the Phase Shift

The phase shift of the sine function \(y = A \sin(Bx - C)\) is \(\frac{C}{B}\). Here, \(C = \frac{\pi}{3}\) and \(B = \frac{1}{3}\), so the phase shift is \(\frac{\frac{\pi}{3}}{\frac{1}{3}} = \pi\). Since this is subtracted, we shift \(\pi\) units to the right.

Step 4: Sketch the Graph

To sketch the graph of \(y = 4 \sin \left(\frac{1}{3}x - \frac{\pi}{3}\right)\), note the following: 1) Amplitude is 4, so the maximum value is 4 and the minimum is -4. 2) The period is \(6\pi\), so the sine wave will complete one full cycle from 0 to \(6\pi\). 3) The phase shift is \(\pi\), start plotting where the sine wave would normally start \(\pi\) units to the right.

Key Concepts

Amplitude

The term amplitude refers to the height or peak of the wave measured from the mid-line or equilibrium of the wave. For sine functions like \(y = A \sin(Bx - C)\), the amplitude is given by the absolute value of \(A\), denoted as \(|A|\). This means that the amplitude does not change with the shift of the wave along the x-axis or changes in period.

In our example, \(y = 4 \sin\left(\frac{1}{3}x - \frac{\pi}{3}\right)\), the amplitude is \(|4|\), which equals 4.

• The wave reaches a maximum height of 4 above the x-axis.
• The wave reaches a minimum height of -4 below the x-axis.

The amplitude determines how "tall" the wave is, but not how "wide" it is. That characteristics depend on the period, which we will explore next.

Period of Sine Function

The period of a sine function is the distance between repetitions of the wave along the x-axis. It tells us how "wide" one complete cycle of the wave is. The formula for the period of the sine function \(y = A \sin(Bx - C)\) is \(\frac{2\pi}{|B|}\). This function stretches or compresses the wave horizontally.

In the function \(y = 4 \sin\left(\frac{1}{3}x - \frac{\pi}{3}\right)\), we use the value \(B = \frac{1}{3}\):

• Calculate the period: \(\frac{2\pi}{\left|\frac{1}{3}\right|} = 6\pi\).
• This means one full cycle of the wave repeats every \(6\pi\) units on the x-axis.

The period affects the wave's horizontal scaling, determining how quickly the wave repeats itself along the x-axis. Now, let's explore how phase shift can affect the starting position of those waves.

Phase Shift

Phase shift describes the horizontal translation of a wave on a graph, which determines where the wave starts. For a sine function of the form \(y = A \sin(Bx - C)\), the phase shift is calculated as \(\frac{C}{B}\).

In the function \(y = 4 \sin\left(\frac{1}{3}x - \frac{\pi}{3}\right)\):

• Phase shift is \(\frac{\frac{\pi}{3}}{\frac{1}{3}} = \pi\).
• The wave is shifted \(\pi\) units to the right.

Phase shift helps you determine where the sine wave will begin oscillating from its usual starting point at the origin. By knowing the phase shift, you can more accurately graph the function by plotting where each cycle starts.