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Chapter 0: Problem 26

Identify the amplitude, period and frequency. $$f(x)=2 \cos 3 x$$

Short Answer

Expert verified
The amplitude of function \(f(x)=2 \cos 3x\) is 2, the period is \(\frac{2\pi}{3}\), and the frequency is 3.

Step by step solution

01

Identify Amplitude

The amplitude of a function is the absolute value of the coefficient of the trigonometric function. Here, the coefficient of the cosine function is 2. So, the amplitude is \(|2|=2\).
02

Identify Period

The period of the function is obtained by dividing \(2\pi\) by the absolute value of the coefficient of \(x\). Here, the coefficient of \(x\) is 3. So, the period is \(\frac{2\pi}{|3|}=\frac{2\pi}{3}\).
03

Identify Frequency

The frequency of the function is the absolute value of the coefficient of \(x\). Here, the coefficient of \(x\) is 3. So, the frequency is \(|3|=3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
Amplitude in trigonometric functions represents how far the function's values can stretch vertically. Think of it like the maximum height of a wave from its middle line to its peak, or down to its trough. An easy way to find the amplitude is by looking at the coefficient in front of the trigonometric function.

For example, in the function \( f(x) = 2 \cos 3x \), the amplitude is determined by the number 2, which is the coefficient of the cosine function. The amplitude is simply the absolute value of this coefficient. Therefore, the amplitude is \( |2| = 2 \).

This means the wave will reach a maximum height of 2 and a minimum of -2 from the midline.
Period
The period of a trigonometric function is the measure of how long it takes for the function to start repeating itself. It's like the length of one complete round of a wave cycle. To find the period of a cosine or sine function, you use the formula \( \frac{2\pi}{|b|} \), where \(b\) is the coefficient of \(x\).

In the function \( f(x) = 2 \cos 3x \), the \(x\) coefficient is 3. So, you calculate the period as \( \frac{2\pi}{|3|} = \frac{2\pi}{3} \).

This means that every \( \frac{2\pi}{3} \) units along the x-axis, the function's values will start over and repeat their pattern. Understanding the period helps predict the behavior of the wave over different intervals.
Frequency
Frequency refers to how often the wave pattern repeats within a specific interval. Typically, it's tied to how many cycles are completed in one unit of time or unit of the horizontal axis. The formula to find frequency is closely related to the period but operates inversely; it is simply the absolute value of the \(x\) coefficient without dividing by \(2\pi\).

In our function \( f(x) = 2 \cos 3x \), the frequency equals \( |3| = 3 \).

This tells us that within any interval of \(2\pi\), the wave will complete 3 full cycles. Knowing the frequency can help determine how quickly the wave oscillates within a given time frame or space.

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