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Amplitude is a key feature of a sinusoidal function that describes how far the function moves above and below its midline. In simple terms, it is the maximum height of the wave from its central axis.
In the given exercise, the amplitude can be determined by looking at the variation in the lung volume, which swings between 2 liters and 4 liters. To find the amplitude, calculate half the difference between the maximum and minimum volume:

• Maximum volume = 4 liters
• Minimum volume = 2 liters
• Amplitude = \( \frac{4 - 2}{2} = 1 \text{ liter} \)

This means that the volume of air in the lungs fluctuates by 1 liter above and below the midline, which is the average point between the maximum and minimum values. Amplitude provides us with information about the range of change around the average, giving us a sense of the dynamics of the sinusoidal pattern.

Frequency in sinusoidal functions reflects how often the cycle of the function repeats over a unit period. It is closely tied to the function's period, as it indicates the number of complete oscillations within a given timeframe.
In the exercise, the person breathes in and out every 3 seconds. This scenario defines the period, and hence influences the frequency of lungs' volume oscillation:

• Given period \( T = 3 \text{ seconds} \)
• Frequency relationship \( T = \frac{2\pi}{b} \)

Thus, solving for frequency "b" gives:

• \( b = \frac{2\pi}{3} \)

This implies that the lungs complete one cycle of breathing within three seconds. The frequency is crucial in determining how rapidly the sinusoidal function completes its cycles, impacting the smooth oscillation portrayed by the function.

The period of a sinusoidal function is the duration it takes to complete one full cycle of its pattern. It is the length of time between identical points on consecutive cycles of the wave.
In this breathing exercise, the period is specified as the time taken to complete one in-and-out breathing cycle:

• Period "T" is \( 3 \text{ seconds} \).

Understanding period helps explain how long the oscillation process takes before repeating its cycle. To establish the period mathematically, use the connection with frequency:

• \( T = \frac{2\pi}{b} \)
• Thus, \( T = 3 \) implies \( b = \frac{2\pi}{3} \)

This tells us that every 3 seconds, the cycle of lung volume oscillation repeats, providing a predictable rhythm. The period is fundamental to charting the progression and timing of the sinusoidal function, enlightening us on the pace at which changes occur.