Problem 89 - Calculus for Biology and Medicine

Question

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\sin x\) and \(y=2 \sin x\)

Step-by-Step Solution

Answer

The graph of \(y = 2 \sin x\) is a vertically stretched version of \(y = \sin x\), with double the amplitude.

Step 1: Understanding the Functions

The first function is the standard sine function, given by \(y = \sin x\). The second function is \(y = 2 \sin x\), which is a vertically stretched version of the sine function.

Step 2: Graphing y = sin x

Graph the function \(y = \sin x\) using the graphing calculator. Notice its properties: it oscillates between -1 and 1 with a period of \(2\pi\) and crosses the x-axis at multiples of \(\pi\).

Step 3: Graphing y = 2 sin x

Graph \(y = 2 \sin x\) on the same axes. This function will also oscillate with the same period \(2\pi\) as \(\sin x\), but its amplitude is increased to 2, meaning it will oscillate between -2 and 2.

Step 4: Comparing the Graphs

Compare both graphs. Although their shapes remain the same and they have the same period, the amplitude of \(y = 2 \sin x\) is twice that of \(y = \sin x\). This results in a 'taller' graph for \(y = 2 \sin x\).

Key Concepts

Understanding the Sine Function

The sine function, represented as \( y = \sin x \), is fundamental in trigonometry. It describes a smooth, periodic wave pattern. The sine wave oscillates smoothly between its maximum and minimum values, giving it its characteristic 'wave' shape. Here are some core aspects of the sine function:

• Range: The values of \( y = \sin x \) range from -1 to 1. This means that the maximum value the sine function can have is 1, and the minimum is -1.
• Zero-crossings: The sine function crosses the x-axis at integer multiples of \( \pi \), such as \( 0, \pi, 2\pi \), and so on. At these points, \( \sin(x) = 0 \).
• Wave shape: It is characterized by its smooth, repetitive nature, showing maximum positive and negative values equidistantly spaced).

Whether you are working with physics, engineering, or trying to understand musical tones, the sine wave is an invaluable tool due to its regular, predictable nature.
Using a graphing calculator, you can visually appreciate how the sine function manifests as a wave.

The Concept of Amplitude

Amplitude refers to the height of the wave from its central axis to its peak. In a trigonometric context:

• Standard amplitude: For the basic sine function \( y = \sin x \), the amplitude is 1. This means the wave moves 1 unit above and below the x-axis.
• Adjusted amplitude: When the function is changed to \( y = 2\sin x \), the amplitude becomes 2. So, the wave now moves 2 units above and below the x-axis.

Changing the amplitude affects the vertical stretch of the wave. A higher amplitude results in a 'taller' wave that reaches higher maxima and lower minima.
Using a graphing calculator to visualize \( y = \sin x \) and \( y = 2\sin x \), you will notice that while the shapes are identical, the graph of \( y = 2\sin x \) appears vertically stretched in comparison to \( y = \sin x \).

Exploring the Period of a Sine Wave

The period of a wave is the length it takes for the wave to complete one full cycle. For \( y = \sin x \), this is a key feature:

• Standard period: For \( y = \sin x \), the period is \(2\pi\). This means the sine wave repeats every \(2\pi\) units along the x-axis.
• Effect of amplitude on period: In the function \( y = 2\sin x \), although the amplitude changes, the period remains the same at \(2\pi\). This is crucial because it shows that vertical stretching affects only the amplitude, not the period.

A graphing calculator can help illustrate the concept of period by allowing you to see these repeating patterns as you engage with different functions such as \( y = \sin x \) and \( y = 2\sin x \).
Recognizing the constant period in varied functions helps immensely in predicting behavior in cyclic phenomena, like sound waves or tides.