Problem 16 - Precalculus: A Concise Course

Question

Find the period and amplitude. $$ y=\frac{5}{2} \cos \frac{x}{4} $$

Step-by-Step Solution

Answer

The amplitude of the function \(y = \frac{5}{2} \cos \frac{x}{4}\) is 2.5 and the period is \(8\pi\).

Step 1 - Identify the amplitude A

From the function \(y = \frac{5}{2} \cos \frac{x}{4}\) we can see that the coefficient of \(\cos\) is \(\frac{5}{2}\). Therefore, amplitude A is \(\frac{5}{2}\) or 2.5.

Step 2 - Identify B for Period Calculation

In the given function \(y = \frac{5}{2} \cos \frac{x}{4}\), the coefficient of \(x\) in \(\cos\) is \(\frac{1}{4}\). So B is \(\frac{1}{4}\).

Step 3 - Calculate the Period

The formula for the period of a cosine function is \(T = \frac{2\pi}{|B|}\). So substituting B with \(\frac{1}{4}\), we get \(T = \frac{2\pi}{|\frac{1}{4}|} = 8\pi\).\nThus, the period of the function is \(8\pi\).

Key Concepts

Amplitude

When exploring the world of trigonometric functions, one important property to grasp is the amplitude. In the context of a cosine function like \(y = \frac{5}{2} \cos \frac{x}{4}\), the amplitude is the distance from the midline to the peak (or trough) of the function. Put simply, it tells us how "tall" or "short" the waves of the function appear.

• In our function, the coefficient in front of the cosine, \(\frac{5}{2}\), determines the amplitude.
• This value of \(\frac{5}{2}\) means that each wave of the cosine graph will rise 2.5 units above or fall 2.5 units below the centerline.

Understanding amplitude helps in visualizing the graph and is crucial in physics and engineering fields where wave behavior is studied.

Period

The period of a function describes how long it takes for the function to repeat itself, cycling through all its values before starting over again. For trigonometric functions, determining the period will show how "stretched" or "compressed" the graph is horizontally.

In our equation \(y = \frac{5}{2} \cos \frac{x}{4}\), the period is calculated using the formula:

• \(T = \frac{2\pi}{|B|}\) where \(B\) is the coefficient of \(x\) inside the cosine.
• Substitute \(B\) with \(\frac{1}{4}\) to find the period as \(T = \frac{2\pi}{|\frac{1}{4}|} = 8\pi\).

This tells us the graph completes a full cycle every \(8\pi\) units along the x-axis. Once you know the period, plotting or analyzing repeating behaviors in a variety of scientific and mathematical contexts becomes more straightforward.

Cosine Function

The cosine function is one of the primary trigonometric functions used across mathematics, physics, and engineering. Represented with the notation \(\cos(x)\), this function describes the cosine values of angles and possesses wave-like properties.

• In our case with \(y = \frac{5}{2} \cos \frac{x}{4}\), the cosine function is vertically stretched by the amplitude (\(\frac{5}{2}\)).
• Its period, altered by the coefficient inside the cosine, shows how quickly it oscillates.

The cosine function is part of the fundamental building blocks when studying periodic phenomena, whether in simple harmonic motion, sound waves, or electromagnetic waves. By understanding its attributes like amplitude and period, you'll gain insights into designing and predicting behaviors in circular and repetitive systems.