Problem 92 - 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 \(y=\sin (2 x)\)

Step-by-Step Solution

Answer

Function \(y=\sin(2x)\) completes two cycles within the period of one cycle of \(y=\sin x\), showing a higher frequency.

Step 1: Understanding the functions

We are given two trigonometric functions: \(y = \sin x\) and \(y = \sin(2x)\). These functions are sinusoidal, and we need to compare them using their respective graphs.

Step 2: Identifying key characteristics

The function \(y = \sin x\) has a period of \(2\pi\), meaning the curve completes one full cycle over the interval from 0 to \(2\pi\). Conversely, \(y = \sin(2x)\) represents a sine wave where the input angle is doubled, affecting its cycle and period.

Step 3: Analyzing the period and amplitude differences

The period of \(y = \sin(2x)\) is shorter than that of \(y = \sin x\). Specifically, by using the formula for the period of \(y = \sin(bx)\), which is \(\frac{2\pi}{|b|}\), we find the period of \(y = \sin(2x)\) is \(\pi\). The amplitude remains the same, as the coefficient of sine (which is 1) hasn't changed.

Step 4: Graphing the functions

Using a graphing calculator, plot both functions \(y = \sin x\) and \(y = \sin(2x)\). You'll observe that \(y = \sin x\) completes one full wave between 0 and \(2\pi\), while \(y = \sin(2x)\) completes two full waves within the same interval, illustrating the increased frequency.

Step 5: Comparison and observations

From the graphs, you should notice that \(y = \sin(2x)\) oscillates twice as quickly as \(y = \sin x\). The period for \(y = \sin x\) is \(2\pi\), and for \(y = \sin(2x)\) is \(\pi\), indicating that doubling the input frequency reduces the period by half.

Key Concepts

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing mathematical functions, especially when dealing with trigonometric functions like sine and cosine. By plotting functions, you can easily compare them and better understand their differences and similarities. When you input a function into a graphing calculator, it converts the equation into a visual graph. This method is particularly helpful for trigonometric functions, where visualizing the wave-like behavior can underline differences that aren't immediately clear from the equation alone.

• Graphing calculators help you plot complex functions quickly.
• You can compare two functions simultaneously to see differences in period and amplitude.
• It's especially useful for visual learners, offering a more intuitive way to grasp concepts.

By seeing the graphs of \(y = \sin x\) and \(y = \sin (2x)\), students can visually appreciate the difference in periods and get insights into the effects of modifying the function's argument.

Period of a Function

The period of a trigonometric function, such as the sine function, is an essential concept in understanding how the function behaves over time. The period is the interval after which the function values repeat. For the basic sine function \(y = \sin x\), the period is \(2\pi\). This means that every \(2\pi\) units along the x-axis, the sine wave starts a new cycle.Modifying the function, such as in \(y = \sin(2x)\), changes the period. The formula for finding the period of \(y = \sin(bx)\) is \(\frac{2\pi}{|b|}\).

• For \(y = \sin x\), the period is \(2\pi\).
• For \(y = \sin(2x)\), the period is halved to \(\pi\), showing the function completes two cycles within the original period of \(y = \sin x\).
• This concept demonstrates how changing the coefficient of the variable affects the function's frequency.

Recognizing period changes helps in predicting the behavior of trigonometric functions, aiding in practical applications and understanding wave phenomena.

Amplitude in Trigonometry

Amplitude refers to the peak value of a wave from its central position, representing how much a wave oscillates from its mean position. In trigonometry, the amplitude tells you how 'tall' the wave appears on a graph. For the sine function \(y = \sin x\), the amplitude is 1, which is derived from the coefficient in front of the sine. The function oscillates between -1 and 1 on the y-axis.In both functions given, \(y = \sin x\) and \(y = \sin(2x)\), the amplitude remains unchanged because the sine coefficient remains 1 in both functions.

• The amplitude does not change unless the coefficient of the sine function changes.
• Regardless of frequency or period changes, amplitude remains dictated by the initial function's coefficient.
• This constancy shows amplitude is independent of input modifications like \(bx\).

Understanding amplitude is vital for interpreting wave behavior and signal strengths in diverse fields such as physics and engineering.