Problem 28 - Thomas Calculus

Question

In Exercises 5–30, determine an appropriate viewing window for the given function and use it to display its graph. $$ y=\frac{1}{10} \sin \left(\frac{x}{10}\right) $$

Step-by-Step Solution

Answer

Set x from 0 to 20π and y from -0.2 to 0.2 for a full cycle view of the graph.

Step 1: Examine the Function

The given function is \( y = \frac{1}{10} \sin \left( \frac{x}{10} \right) \). This is a modified sine function, where the amplitude is scaled down by \( \frac{1}{10} \) and the frequency is affected by \( \frac{x}{10} \).

Step 2: Determine Amplitude and Period

The amplitude is \( \frac{1}{10} \), meaning the wave oscillates between \( -\frac{1}{10} \) and \( \frac{1}{10} \). To find the period, calculate \( \frac{2\pi}{\frac{1}{10}} = 20\pi \). This means the function completes one full cycle over an interval of \( 20\pi \).

Step 3: Choose a Range for x-axis

Since the period is \( 20\pi \), it makes sense to view at least one full cycle. Choose an \( x \)-axis range from \( 0 \) to \( 20\pi \) to capture the entire cycle of the sine wave.

Step 4: Choose a Range for y-axis

The sine function modulated by \( \frac{1}{10} \) oscillates between \( -0.1 \) and \( 0.1 \). Set the \( y \)-axis from \( -0.2 \) to \( 0.2 \) to comfortably fit the graph.

Step 5: Graph the Function

Using a graphing calculator or software, set the viewing window with \( x \) from \( 0 \) to \( 20\pi \) and \( y \) from \( -0.2 \) to \( 0.2 \). Plot the function \( y = \frac{1}{10} \sin \left( \frac{x}{10} \right) \) to observe its behavior.

Key Concepts

Understanding the Sine Function

The sine function is one of the key building blocks in trigonometry. It is represented as \( y = \sin(x) \). In this context, the function shows oscillating behavior that repeats over specific intervals. The shape of a sine wave is smooth and periodic. This means it has a regular repeating pattern.
Sine functions are regularly used to model waves and circular movements, among other things. In this exercise, we examine a modified version of the sine function. This showcases how altering certain components affects its graph. By understanding the core sine function, you can easily grasp changes when you adjust parameters like amplitude or frequency.

Amplitude in the Sine Function

Amplitude is like the height of a wave from its central axis. For the function \( y = \frac{1}{10} \sin(\frac{x}{10}) \), the amplitude is \( \frac{1}{10} \). This means that the wave reaches a maximum height of \( 0.1 \) and a minimum of \( -0.1 \).

• Amplitude determines how tall or short the peaks of the wave appear.
• A larger amplitude results in a taller wave, while a smaller amplitude presents a flatter wave.
• In this case, because the amplitude is \( \frac{1}{10} \), the wave oscillates very gently around the x-axis.

Understanding amplitude helps you anticipate how extreme the values of the wave can get.

Period of the Sine Function

The period of a sine function indicates the length of one complete cycle of the wave.
For \( y = \frac{1}{10} \sin(\frac{x}{10}) \), the period is calculated with the formula:
\ Period = \frac{2\pi}{b} \, where \( b = \frac{1}{10} \).
Thus, the period becomes \( 20\pi \).

• The period signifies that it takes an interval of \( 20\pi \) for the sine wave to repeat its pattern.
• Knowing the period of a sine function helps set up your graph effectively, so you have a complete view of the wave pattern.
• A shorter period results in more frequent oscillations, while a longer period means a more stretched-out wave.

Recognizing the period aids in understanding the frequency and spacing of the wave's cycles.

Determining the Viewing Window

The viewing window in graphing refers to the portion of the graph you choose to display. To effectively graph \( y = \frac{1}{10} \sin(\frac{x}{10}) \), an adequate viewing window should be selected.

• For the x-axis, a range from \( 0 \) to \( 20\pi \) ensures you capture at least one full cycle of the sine wave.
• The y-axis should stretch slightly beyond the maximum and minimum values of the wave. Here, extending from \( -0.2 \) to \( 0.2 \) works well.
• Choosing the right viewing window helps in observing the complete behavior of the function.

An appropriate viewing window reveals the function's oscillations clearly and makes it easier to analyze its properties. Proper setup here enhances the visualization of key features like amplitude and period in the graph.