Problem 98 - Calculus for Biology and Medicine

Question

Find the amplitude and the period of \(f(x)\) $$ f(x)=-\frac{3}{2} \sin \left(\frac{\pi}{3} x\right), \quad x \in \mathbf{R} $$

Step-by-Step Solution

Answer

Amplitude: \(\frac{3}{2}\), Period: 6.

Step 1: Identify the General Form of the Sine Function

The given function is \( f(x) = -\frac{3}{2} \sin \left( \frac{\pi}{3} x \right) \). The general form of a sine function is \( f(x) = A \sin(Bx + C) + D \), where \(A\) is the amplitude, and the period is given by \( \frac{2\pi}{|B|} \).

Step 2: Determine the Amplitude

From the given function, \( f(x) = -\frac{3}{2} \sin \left( \frac{\pi}{3} x \right) \), the coefficient of \( \sin() \) is \( -\frac{3}{2} \). The amplitude is the absolute value of this coefficient, so the amplitude is \( | -\frac{3}{2} | = \frac{3}{2} \).

Step 3: Calculate the Period

The function \( f(x) = -\frac{3}{2} \sin \left( \frac{\pi}{3} x \right) \) has a \( B \) value of \( \frac{\pi}{3} \). The period is calculated using the formula \( \frac{2\pi}{|B|} \), so the period is \( \frac{2\pi}{\frac{\pi}{3}} \). This simplifies to \( \frac{2\pi \cdot 3}{\pi} = 6 \).

Key Concepts

Amplitude

In a trigonometric function like the sine function, the amplitude represents the peak value the wave reaches above or below its central axis. It tells us how "tall" the wave is from its midpoint.
Looking at the function \( f(x) = -\frac{3}{2} \sin \left( \frac{\pi}{3} x \right) \), the amplitude is found by taking the absolute value of the coefficient in front of \( \sin \). This means we focus on \(-\frac{3}{2}\). The negative sign indicates the wave is flipped or "reflected" around the horizontal axis, but it doesn't affect the wave's height.
Therefore, the amplitude is

• \( | -\frac{3}{2} | = \frac{3}{2} \)

An amplitude of \(\frac{3}{2}\) tells us that this wave stretches 1.5 units above and below its center line of zero.

Period

The period of a sine function is the distance along the x-axis it takes for the wave to complete one full cycle. It's like finding out how far you have to travel before the pattern starts repeating again.
In our equation \( f(x) = -\frac{3}{2} \sin \left( \frac{\pi}{3} x \right) \), the term inside the sine function affects this repetition rate. It's called the frequency factor \(B\).
To find the period, you use the formula

• Period = \( \frac{2\pi}{|B|} \)

Here, \(B = \frac{\pi}{3}\). Plugging it into the formula gives:

• Period = \( \frac{2\pi}{\frac{\pi}{3}} = 6 \)

This result tells you that every 6 units along the x-axis, the sine curve begins a new wave.

Sine Function

The sine function is one of the fundamental trigonometric functions, known for its wave-like patterns. It is periodic, which means it repeats at regular intervals, making it useful in modeling cyclical phenomena like sound and light waves.
A general sine function takes the form:

• \( f(x) = A \sin(Bx + C) + D \)

Where:

• \(A\) represents the amplitude, impacting the height of the wave,
• \(B\) sets the period, controlling the frequency of the cycles,
• \(C\) shifts the wave left or right (known as phase shift),
• \(D\) lifts or lowers the entire wave up or down.

In our specific example, there's no horizontal or vertical shift since both \(C\) and \(D\) are zeros, simplifying our focus to just amplitude and period. The sine function is valuable due to its simplicity and its ability to model many natural phenomena.