Problem 66 - Precalculus Mathematics for Calculus

Question

Graph the three functions on a common screen. How are the graphs related? $$y=\sin 2 \pi x, \quad y=-\sin 2 \pi x, \quad y=\sin 2 \pi x \sin 10 \pi x$$

Step-by-Step Solution

Answer

The first two graphs are reflections, while the third involves modulation.

Step 1: Understand the Functions

We have three functions given: \( y=\sin 2 \pi x \), \( y=-\sin 2 \pi x \), and \( y=\sin 2 \pi x \sin 10 \pi x \). The first two are simple sine and negative sine waves, while the third is a product of two sine functions.

Step 2: Graph Each Function

Start by graphing each function on the same set of axes. The function \( y=\sin 2\pi x \) is a standard sine wave with a period of 1. The function \( y=-\sin 2\pi x \) is the reflection of \( y=\sin 2\pi x \) across the x-axis. The function \( y=\sin 2 \pi x \sin 10\pi x \) appears more complex, exhibiting faster oscillations due to the product of the sine waves.

Step 3: Analyze the Graphs

Upon graphing, you'll notice that \( y=\sin 2\pi x \) and \( y=-\sin 2\pi x \) are vertically mirrored. The graph of \( y=\sin 2 \pi x \sin 10 \pi x \) shows a main oscillation pattern of \( y=\sin 2 \pi x \) with additional rapid oscillations caused by the product with \( \sin 10 \pi x \). This creates a wave-like modulation effect.

Step 4: Determine the Relationship

The graphs illustrate that \( y=-\sin 2\pi x \) is the negative version of \( y=\sin 2\pi x \), reflected over the x-axis. The third graph is a modulation of the first sine function by a higher-frequency sine, forming a variable amplitude wave.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, often introduced as part of the unit circle. It represents a periodic wave that oscillates between -1 and 1, described by the equation \( y = \sin x \). For the function \( y = \sin 2\pi x \), the period is slightly altered by the coefficient of \( x \).

• Amplitude: The maximum height of the wave from the center line. For \( \sin x \), this amplitude is 1.
• Period: The interval after which the wave repeats. In \( \sin 2\pi x \), the period is 1, meaning the wave completes one cycle from 0 to 1 on the x-axis.
• Frequency: Number of cycles per unit interval. Here, the frequency is 2\pi, or exactly 1 cycle per unit.

The sine function is crucial in describing oscillatory phenomena like sound and light waves, offering insights into their behavior.

Reflection Across X-Axis

A reflection across the x-axis involves flipping a graph over the x-axis, producing a mirror image. This transformation is seen when we multiply the function by -1, as in \( y = -\sin 2\pi x \).

• Graphical Effect: Each y-value of the original sine function is multiplied by -1, creating an upside-down wave.
• Behavioral Change: The maximum and minimum points of the wave switch positions, affecting its peaks and troughs.
• Visual Symmetry: The original and reflected functions are symmetric with respect to the x-axis, indicating balance in oscillation patterns.

Understanding reflections can help in predicting the behavior of waves, particularly in physics and engineering, where wave properties frequently reflect underlying phenomena.

Product of Sine Functions

The product of two sine functions, such as \( y = \sin 2\pi x \sin 10\pi x \), introduces an intriguing modulation effect. Here, the product leads to a sine-like wave with varying amplitude.

• Modulation: One sine wave (\( \sin 2\pi x \)) provides the base oscillation, while the second (\( \sin 10\pi x \)) adds rapid changes.
• Complexity: The resultant wave has peaks and troughs modulated in size due to the interaction between the two waves.
• Practical Applications: This concept is pivotal in areas like signal processing, where it's used to simulate or analyze complex wave patterns.

By studying such products of sine functions, one gains deeper insight into the dynamics of modulated waves, enriching the understanding of both simple and advanced oscillatory systems.