Problem 30 - Algebra and Trigonometry

Question

\(29-42\) . 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 \(\frac{\pi}{3}\) to the right.

Step 1: Understanding the Equation

The function we have is \( y = 2 \sin\left(x - \frac{\pi}{3}\right) \). This is a sine function of the form \( y = A \sin(Bx - C) \), where \( A \) is the amplitude, \( B \) affects the period, and \( C \) affects the phase shift.

Step 1: Find the Amplitude

The amplitude \( A \) is the coefficient of the sine function. In \( y = 2 \sin\left(x - \frac{\pi}{3}\right) \), \( A = 2 \). Therefore, the amplitude is 2.

Step 2: Find the Period

The period \( T \) of the sine function is given by the formula \( T = \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( x \) inside the sine function. Here, \( B = 1 \). So, the period \( T = \frac{2\pi}{1} = 2\pi \).

Step 3: Find the Phase Shift

The phase shift is determined by \( \frac{C}{B} \) from \( y = A \sin(Bx - C) \). Here, \( C = \frac{\pi}{3} \) and \( B = 1 \). So, the phase shift \( \frac{\pi}{3} / 1 = \frac{\pi}{3} \). This means the graph shifts to the right by \( \frac{\pi}{3} \).

Step 4: Sketching the Graph

To sketch one complete period, start right after the phase shift at \( x = \frac{\pi}{3} \) and go until \( x = \frac{\pi}{3} + 2\pi \). The sine curve will reach its maximum at \( y = 2 \) and its minimum at \( y = -2 \). The complete wave oscillates between those extremes over this interval.

Key Concepts

Amplitude of the Sine Function

When studying a sine function, one of the key features we look out for is the amplitude. This lesson applies even when looking at the function \( y = 2 \sin\left(x - \frac{\pi}{3}\right) \). The amplitude is a measure of how much the wave oscillates above and below its average value, which is typically zero in the case of standard sine functions.

• Definition: In a sine function of the form \( y = A \sin(Bx - C) \), the amplitude is the absolute value of \( A \).
• For the example function, the coefficient \( A = 2 \). Hence, the amplitude is 2.
• This means the sine wave will reach a maximum height of 2 units and a minimum of -2 units from the horizontal axis.
• The wave extends equally above and below the centerline, making this characteristic easily identifiable in a graph.

By identifying the amplitude, we can understand how the sine function stretches or shrinks vertically, allowing us to visualize its peaks and valleys effectively.

Period of the Sine Function

The period of a sine function describes how long it takes for the wave to complete one full cycle. For the function \( y = 2 \sin(x - \frac{\pi}{3}) \), recognizing the period involves looking at the coefficient in front of the variable \( x \).

• Definition: The period \( T \) of a sine wave \( y = A \sin(Bx - C) \) is calculated using the formula \( T = \frac{2\pi}{|B|} \).
• In this case, \( B = 1 \). So, substituting into the formula gives \( T = \frac{2\pi}{1} = 2\pi \).
• This tells us the wave repeats every \( 2\pi \) units in the horizontal direction.
• A longer period would indicate a more stretched out wave, while a shorter period shows it is more compact.

Through understanding the period, we comprehend how frequently the oscillations occur, an essential aspect for graphing sine functions accurately.

Phase Shift of the Sine Function

Phase shift in a sine function reveals how much the graph is horizontally shifted from its standard position. This concept is crucial for drawing and interpreting the sine wave accurately for the function \( y = 2 \sin(x - \frac{\pi}{3}) \).

• Definition: The phase shift in \( y = A \sin(Bx - C) \) can be found using \( \frac{C}{B} \), where the term \( C \) determines the horizontal shift.
• In our equation, \( C = \frac{\pi}{3} \) and \( B = 1 \), resulting in a phase shift of \( \frac{\pi}{3} \).
• This means the entire sine graph moves \( \frac{\pi}{3} \) units to the right.
• The phase shift tells us where the sine wave's critical points move along the \( x \)-axis, like starting points and points of maximum and minimum values.

Mastering the concept of phase shift allows us to accurately predict where the sine cycles start in relation to the origin, ensuring the graph reflects all transformations of the function.