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Chapter 0: Problem 32

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

Short Answer

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

Step by step solution

01

Identify the amplitude

The amplitude of a function in the form \(A \cos(Bx)\) is given by the absolute value of A. Here, A = -2. The amplitude is therefore \(|-2|\), which is 2.
02

Determine period and frequency

The period of the function in the form \(A \cos(Bx)\) is calculated using the formula \(2 \pi / B\). Here, B = 3. The period is therefore \(2 \pi / 3\). The frequency is the reciprocal of the period. Therefore, the frequency is \(1 / (2 \pi / 3) = 3 / 2 \pi\).

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Key Concepts

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

Amplitude
The concept of amplitude is crucial when analyzing trigonometric functions like cosine and sine. Imagine the graph of the function as a wave. The amplitude represents the wave's height from the centerline to its peak.
In mathematical terms, it measures how stretched or compressed a wave is vertically.
For a function in the form of \(f(x) = A \cos(Bx)\), the amplitude is the absolute value of \(A\).
This means you disregard any negative signs to find the positive height of the wave.
  • In the exercise provided, the function is \(f(x) = -2 \cos 3x\).
  • The value of \(A\) is \(-2\), but the amplitude is \(\left|-2\right| = 2\).
This shows how amplitude reflects only the magnitude of the wave's height, not its direction.
Period
If you've ever watched the second hand of a clock, you've observed periodic motion. The period in trigonometric functions is similar: it tells you how long it takes for the function to complete one full cycle.
For functions like \(\cos(Bx)\), the period is determined using the formula: \(\frac{2\pi}{B}\).
This formula explains the horizontal stretch or compression of the wave.
  • In our case with \(f(x) = -2\cos 3x\), \(B\) is equal to 3.
  • The period would then be \(\frac{2\pi}{3}\).
This period means that every \(\frac{2\pi}{3}\) units on the x-axis, the wave pattern begins to repeat itself.Having a grasp on the period helps you understand the timing and spacing of waves, which is invaluable in applications like signal processing and acoustic engineering.
Frequency
Frequency is like the rhythm of the wave, indicating how often it repeats in a given interval. While the period tells us the duration of one cycle, the frequency focuses on the number of cycles that occur in an interval.
The relationship between period and frequency is reciprocal: if you have one, you can find the other.
The formula to find frequency when you have the period is \(\frac{1}{\text{period}}\).
  • In the function \(f(x) = -2\cos 3x\), the period is \(\frac{2\pi}{3}\).
  • Thus, the frequency is \(\frac{1}{\frac{2\pi}{3}} = \frac{3}{2\pi}\).
Frequency is especially beneficial when dealing with waves and oscillations, as it describes how frequently the waves repeat over time. Understanding frequency is essential in fields like music, physics, and any area where vibrating systems are involved.

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