Problem 41 - Precalculus: A Concise Course

Question

Sketch the graph of the function. (Include two full periods.) $$ y=\frac{1}{3} \cos x $$

Step-by-Step Solution

Answer

The graph of \( y=\frac{1}{3} \cos x \) will look like a normal cosine function but will have an amplitude of \( \frac{1}{3} \), and the period of \( 2\pi \). It will pass through the points (0, \( \frac{1}{3} \)), (\( \frac{1}{2}\pi \), 0), (\(\pi \), -\( \frac{1}{3} \)), (\( \frac{3}{2}\pi \), 0), and (\(2\pi \), \( \frac{1}{3} \)) in one cycle with no phase shift. Repeat the pattern for the second period.

Step 1: Identify the Amplitude, Period, and Phase Shift

The amplitude of the function is the absolute value of the coefficient in front of the cosine function, in this case \( \frac{1}{3} \). So, the amplitude is \( \frac{1}{3} \). The period of a basic cosine function is \( 2\pi \) and in this function, there is no coefficient with x, so period remains \( 2\pi \). The phase shift in this function is 0, as there's no addition or subtraction inside the 'cos'.

Step 2: Plot Key Points

Plot the key points for the standard cosine function, starting from 0. Key points include: (0, amplitude), (\( \frac{1}{4} \) of period, 0), (\( \frac{1}{2} \) of period, -amplitude), (3 * \( \frac{1}{4} \) of period, 0), and (period, amplitude). For this function: (0, \( \frac{1}{3} \)), (\( \frac{1}{2}\pi \), 0), (\(\pi \), -\( \frac{1}{3} \)), (\( \frac{3}{2}\pi \), 0), and (\(2\pi \), \( \frac{1}{3} \)).

Step 3: Sketch the Graph

Draw the function by connecting the key points plotted in the previous step, keeping in mind the wave like pattern of the cosine function. Remember to include two full periods.

Key Concepts

Amplitude

When graphing trigonometric functions like the cosine function, one of the first things to determine is the amplitude. The amplitude describes how "tall" or "short" the wave of the graph will be. It is the absolute value of the coefficient in front of the cosine function. For the function given by y = \( \frac{1}{3} \cos x \), this coefficient is \( \frac{1}{3} \). Therefore, the amplitude of this function is \( \frac{1}{3} \).

• This means the maximum height of the graph from the midline is \( \frac{1}{3} \).


• Similarly, the graph will dip \( \frac{1}{3} \) units below the midline.

Calculating the amplitude is essential as it allows us to plot and understand how the graph fluctuates above and below its central axis.

Period of Trigonometric Functions

The period of a trigonometric function like cosine is the distance required for the function to complete one full cycle. For the basic cosine function y = \( \cos x \), the period is \( 2\pi \), meaning the wave-like pattern repeats every \( 2\pi \) units of x.

• If there were a coefficient multiplying x, it would affect the period by compressing or stretching it.


• In general, the formula for the period of a cosine function is \( \frac{2\pi}{b} \), where \(b\) is the coefficient of \(x\).

In the exercise, y = \( \frac{1}{3} \cos x \), there is no coefficient affecting \(x\) so the period remains \( 2\pi \). Understanding the period is key as it allows us to define the interval we need to graph and ensures the function replicates correctly.

Cosine Function

The cosine function is a fundamental trigonometric function that produces a wave-like graph known for its smooth, repetitive oscillations. It starts at its maximum, descends to a minimum, and returns to the maximum, completing one cycle.
To graph a cosine function like y = \( \frac{1}{3} \cos x \), one should:

• Identify key points: The standard key points for a cosine function at the start, middle, and end of a period make sketching easier. For example, these include where it starts at maximum, crosses the midline, reaches a minimum, crosses the midline again, and returns to the maximum.


• Consider transformations: Unlike transformations such as vertical shifts or phase shifts, this function's midline is the x-axis and doesn't have horizontal shifts.

Plot these points carefully to ensure the graph reflects the correct number of periods, oscillating both above and below the midline according to the amplitude of the function.