Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the period and amplitude. $$y=3 \sin 2 x$$

Step-by-Step Solution

Answer

The amplitude of the function \( y=3 \sin 2x \) is 3, and the period is \( \pi \)

Step 1: Identify A and B

The form of the given function is \( y = A \sin(Bx) \). Comparing this with the given function \( y = 3 \sin(2x) \), it can be seen that \( A = 3 \) and \( B = 2 \)

Step 2: Calculate the amplitude

The amplitude of the function is given by the absolute value of \( A \). Therefore, the amplitude is \( |3| = 3 \)

Step 3: Calculate the period

The period of the function is given by \( 2\pi/B \). In this case, \( B = 2 \), so the period is \( 2\pi/2 = \pi \)

Key Concepts

Amplitude

The amplitude of a trigonometric function is a measure of how far the wave extends from its central axis to its peak or trough. It is only applicable to periodic functions like sine and cosine. For the sine function, which is typically written as \( y = A \sin(Bx) \), the amplitude is determined by the coefficient \( A \). The absolute value of \( A \) gives us the amplitude of the sine function, indicating its maximum vertical displacement from the horizontal centerline.

In the exercise, we have the function \( y = 3 \sin(2x) \). Here, the coefficient \( A \) is 3, meaning the amplitude is \( |3| = 3 \). This tells us that the sine wave reaches 3 units above and 3 units below its central axis.

Understanding amplitude is essential as it describes the intensity or strength of the wave. In practical terms, the amplitude of a sound wave affects its loudness, while in a light wave, it influences the brightness.

Period of a Function

The period of a trigonometric function is the length of one complete cycle of the wave before it repeats. For the sine function, the period is traditionally \( 2\pi \) considering a full wave on the trigonometric circle. However, when we manipulate the function, particularly the frequency or the horizontal stretch/compression, the period alters.

In the expression \( y = A \sin(Bx) \), the period can be calculated using \( \frac{2\pi}{B} \). The factor \( B \) affects how quickly the function cycles through its repeats. Thus:

• Higher values of \( B \) shorten the period, causing more cycles to occur in the same space.
• Lower values of \( B \) expand the period, causing fewer cycles.

In the example \( y = 3 \sin(2x) \), the period calculation is \( \frac{2\pi}{2} = \pi \). This means it takes \( \pi \) units along the x-axis for the function to complete one full wave pattern.

Sine Function

The sine function is a fundamental trigonometric function that oscillates up and down in a smooth, repetitive pattern. It is often represented as \( y = A \sin(Bx + C) + D \), but importantly, in its basic form, it is \( y = \sin(x) \) with features you can control by adjusting the parameters of the function.

Key features of the sine function include:

• **Amplitude**: Controlled by the coefficient \( A \).
• **Period**: Modified by the coefficient \( B \), specifically calculated as \( \frac{2\pi}{B} \).
• **Phase Shift**: Altered by \( C \), shifting the function left or right.
• **Vertical Shift**: Adjusted by \( D \), moving the wave up or down the y-axis.

The sine function is integral in various real-world applications, like sound waves and alternating current electricity. Understanding how to manipulate its properties, such as amplitude and period, allows us to fit sine functions to model real phenomena accurately.