Problem 100 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Amplitude

The function given is in the form \( f(x) = A \sin(Bx) \), where \( A = -\frac{3}{2} \). The amplitude of the function \( f(x) \) is the absolute value of \( A \). Calculate the amplitude: \( |A| = |-\frac{3}{2}| = \frac{3}{2} \). Thus, the amplitude is \( \frac{3}{2} \).

Step 2: Calculate the Period

The period of a sine function \( A \sin(Bx) \) is given by the formula \( \frac{2\pi}{|B|} \). From the function \( f(x) = -\frac{3}{2} \sin \left(\frac{\pi}{3} x\right) \), we identify that \( B = \frac{\pi}{3} \). Substitute \( B \) into the period formula: \( \frac{2\pi}{|\frac{\pi}{3}|} = \frac{2\pi}{\frac{\pi}{3}} \). Simplifying this, we find the period to be \( 6 \).

Key Concepts

Amplitude

When talking about the size or height of a wave in a sine function, the amplitude is what we discuss. It's like knowing how tall a wave goes from its central position. In mathematical terms, when you have a trigonometric function like a sine function expressed as \( f(x) = A \sin(Bx) \), the amplitude is the absolute value of the coefficient \( A \). This tells us how much the wave stretches above and below its middle point.
The amplitude of a function like \( f(x) = -\frac{3}{2} \sin\left(\frac{\pi}{3}x\right) \) is \( |A| = \left|-\frac{3}{2}\right| = \frac{3}{2} \). In simple words, no matter how the function dips or rises, the highest point from the middle line is always \( \frac{3}{2} \) units away. Amplitude is always positive, as it signifies a distance.

• The amplitude is the peak height of the wave.
• It is always positive since it's a measure of distance.
• No matter the direction, it remains the same height above or below the central axis.

Understanding amplitude helps in knowing how energetic or intense the wave is.

Period

The period of a sine function explains how long it takes for a wave to complete one full cycle of repeating itself. It's like how many steps it takes to go back to the start on a merry-go-round. For a function in the form of \( A \sin(Bx) \), the period is determined by the formula \( \frac{2\pi}{|B|} \).
In our example, with the sine function \( f(x) = -\frac{3}{2} \sin\left(\frac{\pi}{3} x\right) \), we find \( B = \frac{\pi}{3} \). Therefore, the period is calculated as:

• \( \frac{2\pi}{\left|\frac{\pi}{3}\right|} = \frac{2\pi}{\frac{\pi}{3}} \)
• Simplifying this gives us a period of 6.

This period of 6 tells us that it takes 6 units along the x-axis for the sine wave to start repeating itself. This measurement is crucial in understanding how stretched out or squished the wave cycles are.

Sine Function

The sine function is one of the primary trigonometric functions, defined for any real number. It's represented by \( y = \sin(x) \) and graphically shows a smooth, continuous wave-like pattern that repeats indefinitely. This pattern is fundamental in modeling periodic phenomena.
When working with transformations like in \( f(x) = A \sin(Bx) \), two key characteristics change the basic sine wave:

• The amplitude \( A \) modifies the height of the waves from the center axis.
• The coefficient \( B \) alters the frequency, which determines how quickly the cycles repeat, affecting the period.

The sine function naturally oscillates between -1 and 1 as \( \sin(x) \) over one period from 0 to \( 2\pi \). When transformed with an amplitude \( A \) and period \( \frac{2\pi}{|B|} \), it expands our capacity to fit this versatile wave into numerous real-world contexts, be it sound waves, light patterns, or electricity cycles.