Problem 4 - Applied Calculus

Question

Sketch graphs of the functions. What are their amplitudes and periods? $$y=3 \sin x$$

Step-by-Step Solution

Answer

The amplitude is 3 and the period is \( 2\pi \).

Step 1: Understanding the Sine Function

The function given is \( y = 3 \sin x \). A basic sine function is \( y = \sin x \), which oscillates between -1 and 1 with a period of \( 2\pi \). The graph is a wave that repeats itself every \( 2\pi \) intervals.

Step 2: Determine the Amplitude

The amplitude of a sine function \( y = A \sin x \) is the coefficient \( A \) of the sine. For the function \( y = 3 \sin x \), the amplitude is 3, meaning the wave oscillates between -3 and 3.

Step 3: Determine the Period

The period of a sine function \( y = A \sin(Bx) \) is given by \( \frac{2\pi}{B} \). In this function \( y = 3 \sin x \), \( B = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \). The wave takes \( 2\pi \) units to complete one full cycle.

Step 4: Sketching the Graph

To sketch the graph, plot the wave starting at the origin. At \( x = 0 \), \( y = 0 \). At \( x = \frac{\pi}{2} \), \( y = 3 \). At \( x = \pi \), \( y = 0 \). At \( x = \frac{3\pi}{2} \), \( y = -3 \). And at \( x = 2\pi \), \( y = 0 \). Connect these points smoothly to form the wave for one complete cycle.

Key Concepts

Amplitude

In the realm of trigonometric functions, understanding amplitude is crucial. The amplitude of a sine function, specifically, refers to the height of its peaks from the central axis. For any sine function expressed in the form \( y = A \sin x \), the amplitude is the absolute value of \( A \).

In the exercise, the function is \( y = 3 \sin x \). Here, the amplitude is \( 3 \).

• This means that the wave reaches a maximum height of \( 3 \)
• It also dips to a minimum of \( -3 \)

Hence, for this function, the wave oscillates symmetrically, as it reaches 3 units above and below its central horizontal axis. The amplitude does not affect the wave's frequency or how quickly it oscillates. It solely impacts its height.

Period

The period of a function is like its heartbeat—the duration of one complete cycle. For a standard sine function \( y = \sin x \), this cycle spans \( 2\pi \) units along the x-axis. The period helps determine how "stretched" or "compressed" the wave appears.

If a function is altered to \( y = A \sin(Bx) \), the period is given by \( \frac{2\pi}{B} \). Our function, \( y = 3 \sin x \), has \( B = 1 \), leading to a period of \( \frac{2\pi}{1} = 2\pi \).

• This means each cycle of the wave repeats every \( 2\pi \) units.

Changing \( B \) would modify the frequency and the period of the wave, demonstrating the ihow compressible the wave is along the x-axis. The amplitude, though, remains unaffected by any changes to the period.

Sine Function

The sine function is one of the core functions in trigonometry. The basic form of this function is \( y = \sin x \). It's known for its smooth, oscillating wave pattern, often called a sinusoidal wave.

This wave:

• Starts at the origin \((0,0)\).
• Reaches its highest positive point at \( x = \frac{\pi}{2} \).
• Returns to zero at \( x = \pi \).
• Dives to its lowest negative point at \( x = \frac{3\pi}{2} \).
• Finishes one full cycle at \( x = 2\pi \).

The sine function is periodic and continues indefinitely, repeating its cycle. The given function \( y = 3 \sin x \) maintains the inherent properties of the sine wave but stretches in height because of the amplitude, keeping the classic pattern intact. Changes in the coefficient \( A \) influence its amplitude, while \( B \) affects its period, demonstrating how versatile this function can be when modeling periodic phenomena.