Problem 30 - Precalculus Mathematics for Calculus

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$y=2 \sin \left(x-\frac{\pi}{3}\right)$$

Step-by-Step Solution

Answer

Amplitude is 2, period is \(2\pi\), phase shift is \(\frac{\pi}{3}\) right.

Step 1: Identify the Amplitude

The general form of the sine function is \( y = a \sin(bx - c) + d \). In this function, \( a = 2 \). The amplitude is the absolute value of \( a \), so the amplitude is \( |2| = 2 \).

Step 2: Calculate the Period

The period of a sine function is given by \( \frac{2\pi}{b} \). For the function \( y = 2 \sin(x - \frac{\pi}{3}) \), \( b = 1 \). Therefore, the period is \( \frac{2\pi}{1} = 2\pi \).

Step 3: Determine the Phase Shift

The phase shift is determined by \( \frac{c}{b} \) from the equation \( y = a \sin(bx - c) + d \). Here, \( c = \frac{\pi}{3} \) and \( b = 1 \). Thus, the phase shift is \( \frac{\pi}{3} \) to the right.

Step 4: Graph One Complete Period

To graph one complete period of the function, we start at the phase shift \( \frac{\pi}{3} \) to the right of the origin. The sine function will complete one full cycle from \( \frac{\pi}{3} \) to \( \frac{\pi}{3} + 2\pi \). The significant points we plot for a sine function over one period are maximum at the amplitude (2), minimum at (-2), and x-intercepts.

Key Concepts

Amplitude

In trigonometric functions, amplitude refers to the maximum distance a wave reaches from its equilibrium or center line. For sine and cosine functions, this is determined by the coefficient of the sine or cosine term. For the function given, which is of the form \[ y = a \sin(bx - c) + d \] "\(a\)" represents the amplitude. Thus, the amplitude is the absolute value of this coefficient, since amplitude is a measure of distance and must always be positive.

• For our function \( y = 2 \sin(x - \frac{\pi}{3}) \), "\(a = 2\)".
• The amplitude is thus \(|2| = 2\).

This implies the wave will reach as high as 2 units above the center and as low as 2 units below it. By understanding amplitude, students can better anticipate how 'tall' or 'short' the graph of a wave will appear.

Period

The period of a trigonometric function signifies the distance along the x-axis it takes for the function to complete one full cycle. For sine and cosine functions, this is especially important as it indicates the frequency of the wave.The general way to compute the period of the standard sine or cosine function \( y = a \sin(bx - c) \) is to use the formula \[ \text{Period} = \frac{2\pi}{b} \]

• In our example: \( b = 1 \).
• Substituting this into the formula gives \( \frac{2\pi}{1} = 2\pi \).

This means the wave will restart every \( 2\pi \) units along the x-axis. Understanding the period helps in predicting where the wave will repeat, which is crucial for graphing.

Phase Shift

Phase shift in trigonometric functions refers to the horizontal displacement of the wave, which shows how much the entire graph shifts to the left or right. This shift is influenced by the constant "\( c \)" in the function \[ y = a \sin(bx - c) \]. The phase shift is calculated using:\[ \text{Phase Shift} = \frac{c}{b} \].

• In our instance: \( c = \frac{\pi}{3} \) and \( b = 1 \).
• Thus, the phase shift is \( \frac{\pi}{3} \).

This means the entire sine wave is shifted to the right by \( \frac{\pi}{3} \) units. Predicting the phase shift is pivotal for accurately plotting the starting point of the function on the graph.