Problem 33 - 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: \(\pi\) right.

Step 1: Determine the Amplitude

The amplitude of a sine function given by \(y=a \sin(bx+c)\) is the absolute value of \(a\). In this function, \(a=4\). Therefore, the amplitude is \(|4|=4\).

Step 2: Calculate the Period

The period of a sine function \(y=a \sin(bx+c)\) 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: Find the Phase Shift

The phase shift of the sine function \(y=a \sin(bx+c)\) is calculated by solving \(-\frac{c}{b}\). Here, \(b=\frac{1}{3}\) and \(c=-\frac{\pi}{3}\), thus the phase shift is:\[\frac{-(-\frac{\pi}{3})}{\frac{1}{3}} = \pi.\] This indicates a shift to the right by \(\pi\) units.

Step 4: Sketch the Graph

Start by plotting the basic structure of \(y=\sin(x)\) over one period \([0, 6\pi]\). Scale the amplitude to 4, and shift the graph \(\pi\) units to the right to reflect the phase shift. The wave will repeat every \(6\pi\) due to its period, maintaining maximum and minimum points at \(y=4\) and \(y=-4\) respectively.

Key Concepts

Understanding Amplitude

Amplitude is a fundamental property of trigonometric functions, particularly for sine and cosine. It represents the height from the central axis of the wave to its peak or trough. This means it's how

• high the graph reaches above the midpoint, and
• low it dips below the midpoint.

In the equation given, amplitude is found by looking at the coefficient of the sine function. For the function \[y = a \sin(bx + c)\],the amplitude is \(|a|\). Therefore, if we have \(y = 4 \sin\left(\frac{1}{3}x - \frac{\pi}{3}\right)\),the amplitude is \(|4| = 4\).

This value tells us that our sine wave will achieve a maximum of 4 units above the x-axis and a minimum of 4 units below the x-axis. Understanding amplitude helps in scaling your graph vertically appropriately.

Calculating the Period

The period of a trigonometric function is the distance over which the wave's shape repeats itself.
For sine and cosine functions, the period is closely linked to the coefficient of x.

• It's calculated using the formula \(\frac{2\pi}{|b|}\).

For our specific function \(y = 4 \sin\left(\frac{1}{3}x - \frac{\pi}{3}\right)\),b is \(\frac{1}{3}\). Thus, the period is computed as follows:
\[\frac{2\pi}{\frac{1}{3}} = 6\pi\].
This means the wave repeats every \(6\pi\) units along the x-axis.

Understanding the period is key for correctly sketching the graph, ensuring that the function accurately represents the cyclic nature.

Decoding Phase Shift

Phase shift refers to the horizontal translation of the graph of a trigonometric function. It indicates how far the function's graph shifts left or right from its usual starting position. To find the phase shift,

• use the formula \(-\frac{c}{b}\).

For our equation \(y = 4 \sin\left(\frac{1}{3}x - \frac{\pi}{3}\right)\),we have \(c = -\frac{\pi}{3}\) and \(b = \frac{1}{3}\). When solving, you'll find:
\[\frac{-(-\frac{\pi}{3})}{\frac{1}{3}} = \pi\].This result means the graph is shifted \(\pi\) units to the right.

Phase shifts are crucial for setting the phase or initial position of your trigonometric graph, helping align it with particular data or events.