Problem 18 - Precalculus: A Concise Course

Question

Find the period and amplitude. $$ y=\frac{2}{3} \cos \frac{\pi x}{10} $$

Step-by-Step Solution

Answer

The amplitude of the given function is \(\frac{2}{3}\) and the period is 20.

Step 1: Identify the Amplitude

The amplitude is given by the absolute value of A in the equation \(y = A cos(Bx + C) + D\). Therefore, in the given equation \(y = \frac{2}{3} \cos \frac{\pi x}{10}\), the amplitude is \(| \frac{2}{3} | = \frac{2}{3}\)

Step 2: Identify the Coefficient of x

We need B from the equation \(y = A cos(Bx + C) + D\) to calculate the period of the function. Here, \(B = \frac{\pi}{10}\)

Step 3: Calculate the Period

The period of a cosine function is calculated by \(T = \frac{2\pi}{B}\). Substituting the value of B from step 2, we get \(T = \frac{2\pi}{\frac{\pi}{10}} = 20\)

Key Concepts

Amplitude of Cosine Function

The amplitude of a cosine function is a measure of how far the peaks and troughs of the wave are from the function's horizontal axis. In other words, it describes the height of the wave. For the function \( y = A \textrm{cos}(Bx + C) + D \) the amplitude is represented by the absolute value of the coefficient \( A \). This absolute value ensures the amplitude is non-negative, as the concept of 'height' does not consider direction.

In the exercise \( y = \frac{2}{3} \textrm{cos} \frac{\pi x}{10} \), the coefficient in front of the cosine, \( \frac{2}{3} \), tells us the amplitude is \( \frac{2}{3} \). This means the maximum height above and below the central axis (often the x-axis for standard cosine functions) the curve will reach is \( \frac{2}{3} \) units. Understanding amplitude is pivotal to graphing trigonometric functions since it directly influences the vertical stretch of the wave.

Period of Cosine Function

The period of a cosine function, often denoted as \( T \), is the distance over which the function's shape repeats. For a standard cosine function of the form \( y = A \textrm{cos}(Bx + C) + D \), the period is given by the formula \( T = \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( x \) within the cosine.

In our example, the function provided is \( y = \frac{2}{3} \textrm{cos} \frac{\pi x}{10} \), and the coefficient of \( x \) is \( \frac{\pi}{10} \). Plugging \( B \) into the period formula we get \( T = \frac{2\pi}{\frac{\pi}{10}} = 20 \), meaning the cosine function completes one full cycle over an interval of 20 units along the x-axis. Understanding the period is essential for analyzing the frequency of trigonometric functions, as it tells us how rapidly they oscillate.

Trigonometric Function Transformations

Trigonometric function transformations involve altering the original function \( y = A \textrm{cos}(Bx + C) + D \) to shift, stretch, or reflect its graph on a coordinate plane. These transformations result from changing the values of the constants \( A, B, C, \) and \( D \) within the equation.

• The constant \( A \) affects the amplitude, as discussed earlier.
• \( B \) affects the period of the function, changing how quickly the wave oscillates.
• The constant \( C \) shifts the graph horizontally, which is known as the phase shift.
• Finally, \( D \), if present, shifts the graph vertically, changing the midline of the wave.

Understanding these transformations allows us to graph complicated trigonometric functions by applying sequential modifications to a standard function. For instance, if you encounter a negative value for \( A \), the graph will be reflected across the horizontal axis. When tackling homework problems, recognizing and applying the correct transformations can simplify the graphing process significantly.