Problem 97 - Calculus for Biology and Medicine

Question

Find the amplitude and the period of \(f(x)\) : $$ f(x)=-4 \sin (\pi x), \quad x \in \mathbf{R} $$

Step-by-Step Solution

Answer

Amplitude is 4, and period is 2.

Step 1: Identify the Amplitude

The amplitude of a sine function in the form \( f(x) = a \sin(bx) \) is given by the absolute value of \( a \). In this case, \( a = -4 \), so the amplitude is \( |a| = |-4| = 4 \).

Step 2: Determine the Period

The period of a sine function in the form \( f(x) = a \sin(bx) \) is given by \( \frac{2\pi}{b} \). In this function, \( b = \pi \), so the period is \( \frac{2\pi}{\pi} = 2 \).

Key Concepts

Understanding Amplitude in Sine Functions

Amplitude is a crucial aspect of trigonometric functions like sine and cosine. It represents the height of the wave from its central axis to its peak. In simpler words, amplitude tells us how "tall" the wave is on a graph. For the function \( f(x) = a \sin(bx) \), the amplitude is found by taking the absolute value of \( a \).

• In the function \( f(x) = -4 \sin(\pi x) \), \( a = -4 \).
• Hence, the amplitude is \( |a| = |-4| = 4 \).

It's important to note that the negative sign in \(-4 \) does not affect the amplitude; it simply indicates a reflection across the horizontal axis. Regardless of this reflection, the wave still rises and falls by a height of 4 units.

Exploring Periodicity in Trigonometric Waves

Periodicity refers to how long it takes for a sine wave to complete one full cycle. This is called the period, and it's a measure of the wave's frequency. For the standard function \( f(x) = a \sin(bx) \), the period is given by \( \frac{2\pi}{b} \).

• In this scenario, \( b = \pi \).
• Therefore, the period is \( \frac{2\pi}{\pi} = 2 \).

This means that the sine wave repeats its pattern every 2 units along the x-axis. A shorter period would imply a more frequent wave pattern within the same horizontal span. Understanding periodicity is essential in graphing trigonometric functions accurately and analyzing their behavior over different intervals.

The Characteristics of Sine Waves

The sine wave is one of the fundamental graphs in trigonometry. Characteristically, it is a continuous and smooth wave that oscillates above and below a central axis. The shape and form of a sine wave are determined by its amplitude and period among other attributes.

• The midline, or central axis, is where the wave oscillates about.
• The maximum value is determined by the amplitude, reaching both its peak and trough.
• Each cycle of the wave looks identical, defined by the periodicity.

In the case of \( f(x) = -4 \sin(\pi x) \), the wave is flipped vertically, as seen by the negative amplitude, but it still possesses the classical curvature of sine waves.
Understanding sine wave properties can help in various fields such as physics, engineering, and any discipline involving periodic behavior.