Problem 4 - Applied Calculus

Question

Graph the function. What is the amplitude and period? $$y=3 \sin 2 x$$

Step-by-Step Solution

Answer

Amplitude is 3 and period is \( \pi \).

Step 1: Identify the Standard Form of Sine Function

The standard form for the sine function is \( y = a \sin(bx) + c \), where \( a \) is the amplitude and \( \frac{2\pi}{b} \) is the period of the function.

Step 2: Determine the Amplitude

The amplitude \( a \) is the coefficient in front of the sine function. For \( y = 3 \sin 2x \), the value of \( a \) is 3. Thus, the amplitude is 3.

Step 3: Calculate the Period

The period of a sine function is determined by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the sine function. Here, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).

Step 4: Sketch the Graph of the Function

To sketch \( y = 3 \sin 2x \), plot the function over one period \( [0, \pi] \). At \( x = 0 \), \( y=0 \); at \( x = \frac{\pi}{4} \), \( y = 3 \); at \( x = \frac{\pi}{2} \), \( y = 0 \); at \( x = \frac{3\pi}{4} \), \( y = -3 \); and at \( x = \pi \), \( y = 0 \). Repeat this pattern for additional cycles.

Key Concepts

Amplitude

When it comes to trigonometric graphs, particularly sine waves, the term 'amplitude' refers to the peak value or maximum height of the wave from its central axis. In the context of the sine wave equation, the amplitude is represented by the coefficient in front of the sine function. For example, given the equation \( y = 3 \sin 2x \), the coefficient \( 3 \) is the amplitude.
What does this mean practically? The amplitude determines how tall or short the wave peaks are from the horizontal axis, also known as the axis of symmetry. It measures the height of the crest or depth of the trough from the middle of the wave.

• An amplitude of \( 3 \) means the graph will oscillate 3 units above and below the central axis (usually the x-axis unless otherwise translated).
• Increasing the amplitude results in taller peaks and deeper valleys, while decreasing it makes the graph flatter.

Understanding amplitude helps you predict how a trigonometric function will react in real-life scenarios like sound waves, light waves, or any oscillatory motion.

Period of Trigonometric Functions

The 'period' of a trigonometric function is the magnitude of the horizontal distance over which the function completes one full cycle. For sine functions, the notion of period is a key component of understanding how the graph behaves over time. In the standard form of the sine function \( y = a \sin (bx) + c \), the period is calculated using the formula \( \frac{2\pi}{b} \). This is because the sine function is inherently periodic with a fundamental loop of \( 2\pi \). By introducing the multiplier \( b \) to the variable \( x \), the function compresses or stretches horizontally.

• In the equation \( y = 3 \sin 2x \), the coefficient \( b = 2 \) alters the period: \( \frac{2\pi}{2} = \pi \). This implies that the graph repeats its cycle every \( \pi \) units.
• If \( b > 1 \), the period is shorter, causing the wave to cycle more frequently. Conversely, if \( 0 < b < 1 \), the period is longer, and cycles are less frequent.

Grasping the concept of period is vital for analyzing how trigonometric functions mimic periodic phenomena like tides, seasons, or even simple tasks aligning with specific time intervals.

Graphing Sine Waves

Graphing a sine wave involves plotting the function across its domain, typically one full period or more to capture the wave's repetitive nature. Let’s explore how to graph \( y = 3 \sin 2x \) to better understand this process.To plot the sine function \( y = 3 \sin 2x \):
1. Identify key points in one period \([0, \pi]\): The function starts at the origin (0, 0), reaches a peak (\( \pi/4, 3\)), returns to the x-axis (\( \pi/2, 0\)), hits a trough (\( 3\pi/4, -3\)), and completes the cycle back at the x-axis (\( \pi, 0\)). 2. Mark these points on your graph and smoothly connect them using the familiar wave-like curve. This is the basic shape of one period of your sine wave.

• After plotting the initial period, extend the pattern to the left and right for multiple cycles if needed.
• The amplitude \( 3 \) indicates how high and low the wave rises and falls from the middle line.
• The period \( \pi \) represents the horizontal length before the pattern repeats.

Graphing sine waves effectively illustrates the regular, cyclical patterns of functions, making it easier to interpret various real-world periodic occurrences consistently.