Problem 23 - Precalculus Mathematics for Calculus

Question

Find the amplitude and period of the function, and sketch its graph. $$y=-\frac{1}{3} \cos \frac{1}{3} x$$

Step-by-Step Solution

Answer

Amplitude: \(\frac{1}{3}\), Period: \(6\pi\).

Step 1: Identify Amplitude

The amplitude of a cosine function of the form \( y = a \, ext{cos} \, bx \) is the absolute value of the coefficient \( a \). In this case, \( a = -\frac{1}{3} \), so the amplitude is \( \left| -\frac{1}{3} \right| = \frac{1}{3} \).

Step 2: Calculate Period

The period of a cosine function is given by \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). For the given function \( y = -\frac{1}{3} \, ext{cos} \, \frac{1}{3} x \), \( b = \frac{1}{3} \), so the period is \( \frac{2\pi}{\frac{1}{3}} = 6\pi \).

Step 3: Sketch the Graph

To sketch the graph of \( y = -\frac{1}{3} \, ext{cos} \, \frac{1}{3} x \): 1. Determine the key points for one period from 0 to \( 6\pi \).2. Note the negative sign in the amplitude which reflects the graph over the x-axis.3. Key points occur at \( x = 0, \frac{3\pi}{2}, 3\pi, \frac{9\pi}{2}, 6\pi \) with corresponding y-values: \(-\frac{1}{3}, 0, \frac{1}{3}, 0, -\frac{1}{3}\).4. Plot these points and connect them smoothly, repeating the pattern for additional periods if desired.

Key Concepts

Amplitude

The amplitude of a trigonometric function helps us understand how far its graph stretches vertically from its central axis, usually the x-axis for these types of problems.
Given a cosine function in the general form: \( y = a \cos(bx) \), the amplitude can be found by taking the absolute value of the coefficient \( a \).
For the function \( y = -\frac{1}{3} \cos \frac{1}{3} x \), the coefficient \( a \) is \(-\frac{1}{3}\).

• To find the amplitude, evaluate \( |a| \).
• This results in an amplitude of \( \frac{1}{3} \).

This means that the highest and lowest values of the function are \( \frac{1}{3} \) units above and below the horizontal axis, effectively setting the boundaries for the graph's peak and trough.

Period of Cosine Function

The period of a trigonometric function is the distance between consecutive repeating points on the graph, where the pattern begins to repeat itself.
For cosine functions, the period is calculated using the formula \( \frac{2\pi}{b} \), whereby \( b \) is the frequency which is the coefficient of \( x \) within the cosine function.
In the function \( y = -\frac{1}{3} \cos \frac{1}{3} x \), \( b \) is \( \frac{1}{3} \).

• Inserting \( b \) into the period formula, we get \( \frac{2\pi}{\frac{1}{3}} = 6\pi \).
• Thus, this cosine function has a period of \( 6\pi \), which means after this interval, the function will start repeating its shape.

The period lets us know how wide one cycle of the graph's wave extends along the x-axis before repeating the same sequence of values.

Graph Sketching

Sketching the graph of a trigonometric function involves plotting points for key values within one cycle or period.
Let's use the amplitude and period we determined to help plot the function \( y = -\frac{1}{3} \cos \frac{1}{3} x \).

• First, note the amplitude is \( \frac{1}{3} \), so the graph's maximum and minimum y-values are these distances from the x-axis.
• Due to the negative amplitude, the graph is reflected over the x-axis.

To sketch the graph:

• Determine key points within one period \( [0, 6\pi] \).
• For these points, the x-values include 0, \( \frac{3\pi}{2} \), \( 3\pi \), \( \frac{9\pi}{2} \), and \( 6\pi \).
• The corresponding y-values are \(-\frac{1}{3} \), 0, \( \frac{1}{3} \), 0, and \(-\frac{1}{3} \) respectively due to the cosine wave's pattern.

Plot these points, drawing a smooth curve through each, and extend the pattern over additional periods if needed to get a fuller sense of the function's harmony.