Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a two-period interval. Give the period and amplinde. $$y=-2 \cos 3 x$$

Step-by-Step Solution

Answer

Amplitude: 2, Period: \(\frac{2\pi}{3}\).

Step 1: Identify the amplitude of the function

The amplitude of a trigonometric function of the form \(y = a \cos b x\) is the absolute value of \(a\). In this case, since \(a = -2\), the amplitude is \(|-2| = 2\). Thus, the function will oscillate 2 units above and below the horizontal axis.

Step 2: Determine the period of the function

The period \(T\) of a function \(y = a \cos b x\) can be calculated using the formula \(T = \frac{2\pi}{b}\). Here, \(b = 3\), so the period is \(T = \frac{2\pi}{3}\). This means that the function completes one full cycle every \(\frac{2\pi}{3}\) units.

Step 3: Graph the function over a two-period interval

To graph the function over a two-period interval, start by identifying the x-values that define this interval. Since one period is \(\frac{2\pi}{3}\), two periods would be \(2 \times \frac{2\pi}{3} = \frac{4\pi}{3}\). Begin the graph at \(x = 0\) and extend to \(x = \frac{4\pi}{3}\). Mark key points on the graph at intervals of \(\frac{2\pi}{3}\) to show the full cycles.

Step 4: Plot key points of the first period

Start at \(x = 0\), where \(y = -2\cos(3 \times 0) = -2\cos(0) = -2\). At \(x = \frac{\pi}{6}\), \(y = -2\cos(3 \times \frac{\pi}{6}) = -2\cos(\frac{\pi}{2}) = 0\). At \(x = \frac{\pi}{3}\), \(y = -2\cos(3 \times \frac{\pi}{3}) = -2\cos(\pi) = 2\). At \(x = \frac{\pi}{2}\), \(y = 0\). Finally, at \(x = \frac{2\pi}{3}\), \(y = -2\), completing one cycle.

Step 5: Plot key points of the second period

Continue the pattern from the end of the first period. At \(x = \frac{5\pi}{6}\), \(y = 0\). At \(x = \frac{2\pi}{3} + \frac{\pi}{3} = \pi\), \(y = 2\). At \(x = \frac{3\pi}{2}\), \(y = 0\). Finally, at \(x = \frac{4\pi}{3}\), \(y = -2\) again. These points complete the second cycle.

Step 6: Draw the cosine graph

Connect the plotted points with a smooth wave to form the graph of the function. Note that the graph starts from \(-2\), rises to 0, then to 2, and returns to 0, and finally to \(-2\) over each period.

Key Concepts

Amplitude

When graphing trigonometric functions, amplitude is one of the key features we examine. The amplitude represents how far the graph of the function stretches upwards and downwards starting from its central axis.
In mathematical terms, the amplitude of a function given by the formula \(y = a \cos bx\) is the absolute value of \(a\), denoted as \(|a|\).
For the function \(y = -2 \cos 3x\), the coefficient \(a\) is \(-2\). Therefore, the amplitude is \(|-2| = 2\).
This indicates that the graph will oscillate between 2 units above and 2 units below the horizontal axis, creating a total range of 4 units from peak to peak. Understanding amplitude helps in predicting the overall *height* and *depth* of the wave.

Period of a Function

The period of a trigonometric function is the distance along the x-axis before the function repeats its pattern. It's essentially how long it takes for the function to complete one full cycle.
For the cosine function \(y = a \cos bx\), the period \(T\) is calculated using the formula \(T = \frac{2\pi}{b}\).
In our specific case, \(b = 3\). Therefore, the period of \(y = -2 \cos 3x\) is \(T = \frac{2\pi}{3}\).
This means the function completes one cycle of oscillation every \(\frac{2\pi}{3}\) units.
Recognizing the period allows us to accurately graph and analyze the trigonometric wave, showing where the peaks and troughs recur along the x-axis. It is crucial for understanding the *rate* at which these wave patterns repeat.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often represented as \(y = \cos x\). It is characterized by its wave-like pattern which starts at a maximum value, decreases to a minimum value, and then returns to the maximum. This complete cycle represents one *period*.
The basic cosine function \(y = \cos x\) has several key properties:

• Symmetry: Even function; symmetrical about the y-axis.
• Range: The standard range is from \(-1\) to \(1\).
• Periodicity: The standard period is \(2\pi\).

In the modified function \(y = -2 \cos 3x\), the negative sign reflects the function across the x-axis, starting the cycle from the minimum value (-amplitude) instead of the maximum. The coefficient \(3\) in the argument \(3x\) compresses the period to \(\frac{2\pi}{3}\). The amplitude \(-2\) shows the stretched height of the wave; however, the sign indicates a reflection.
Understanding these modifications helps accurately sketch the cosine curve and interpret its behavior in real-world applications.