Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=-\frac{2}{3} \cos \left(\frac{x}{2}-\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The amplitude of the graph is \(\frac{2}{3}\) and the period is \(4\pi\).

Step 1: Identify the amplitude

The amplitude of a function is the absolute value of the coefficient of the cosine term. Here, \(a = -\frac{2}{3}\), so the amplitude is \(|a| = \frac{2}{3}\).

Step 2: Identify the period

The period of a cosine function is given by the formula \(P = \frac{2\pi}{|b|}\), where b is the coefficient of x in the argument of the function. Here \(b = \frac{1}{2}\), so the period is \(P = \frac{2\pi}{\frac{1}{2}} = 4\pi\).

Step 3: Graph the function

Now graph the function using a graphing utility. The function is cosine, which starts at a high point, goes to a low point, and back up. The graph should show two full periods, which go from x=0 to x=\(8\pi\), with a maximum of \(\frac{2}{3}\) and a minimum of \(-\frac{2}{3}\).

Key Concepts

Amplitude of Trigonometric Functions

When it comes to graphing trigonometric functions, understanding amplitude is key. Amplitude refers to the height from the center line (or axis) of the graph to its peak. This tells us how "tall" the wave is. It influences how much the graph oscillates above and below its central axis.

In a cosine function, such as \(y = a \cos(bx + c)\), the amplitude is the absolute value \(|a|\). For example, in the function \(/y = -\frac{2}{3} \cos\left(\frac{x}{2} - \frac{\pi}{4}\right)\), the amplitude \(|a|\) is calculated by taking the absolute value of the coefficient of \(\cos\), which is \(-\frac{2}{3}\).

- **Step for finding amplitude:** - Look at the coefficient in front of the cosine; in our case, it is \(-\frac{2}{3}\). - Take the absolute value to get the amplitude: \(\left|-\frac{2}{3}\right| = \frac{2}{3}\).

Thus, the amplitude tells us that the graph of this cosine function oscillates between \(\frac{2}{3}\) and \(-\frac{2}{3}\). This range is essential for sketching an accurate graph.

Period of Trigonometric Functions

The period of a trigonometric function has to do with how often the waves repeat. It is the distance required for one complete cycle of the wave on the x-axis.

For a cosine function defined as \(y = a \cos(bx + c)\), the period \(P\) is calculated using the formula \(P = \frac{2\pi}{|b|}\), where \(b\) is the coefficient of \(x\).

In the case of the function \(y = -\frac{2}{3} \cos\left(\frac{x}{2} - \frac{\pi}{4}\right)\), we identify:- The coefficient of \(x\) (or \(b\)) is \(\frac{1}{2}\).- Calculate the period: \(P = \frac{2\pi}{\frac{1}{2}} = 2\pi \times 2 = 4\pi\).> The period \(4\pi\) implies that it takes \(4\pi\) units along the x-axis for the pattern to repeat itself. Thus, the graph will complete its full cycle of peaks and troughs every \(4\pi\) units, making it easy to plot two full periods by going from \(x = 0\) to \(x = 8\pi\).

Understanding the period helps predict the function's behavior and is an invaluable tool for graphing.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and is crucial in various areas of mathematics and engineering. It describes a smooth, oscillating wave that's symmetrical about the y-axis. The general form is \(y = a \cos(bx + c) + d\), where:

• \(a\) indicates the amplitude
• \(b\) affects the period
• \(c\) shifts the graph horizontally
• \(d\) shifts it vertically

In the function \(y = -\frac{2}{3} \cos\left(\frac{x}{2} - \frac{\pi}{4}\right)\):

• Amplitude \(|a| = \frac{2}{3}\), altering the height of peaks and troughs.
• The horizontal shift \(-\frac{\pi}{4}\) means the entire wave is shifted to the right by \(\frac{\pi}{4}\) units.
• The negative sign before the amplitude flips the wave across the x-axis.

This negative sign creates a reflection, meaning the wave starts from a trough instead of a peak. Understanding these components allows you to construct and analyze the graph effectively. It always helps to draw out these transformations step by step for a precise graphing process.