Find the period and amplitude of the function:
[tex]y=-4 \cos \left(\frac{2 \pi}{3} x\right)[/tex]
Give the exact values, not decimal approximations.
Answer (2)
Identify the amplitude as the absolute value of the coefficient of the cosine function: ∣ − 4∣ = 4 .
Identify B as the coefficient of x inside the cosine function: B = 3 2 π .
Calculate the period using the formula T = B 2 π : T = 3 2 π 2 π = 3 .
State the amplitude and period: Amplitude is 4 and the period is 3, so the final answer is A m pl i t u d e = 4 , P er i o d = 3 .
Explanation
Understanding the Problem We are given the function y = − 4 cos ( 3 2 π x ) and asked to find its period and amplitude. Let's break down what each of these terms means and how to find them.
General Form of Cosine Function The general form of a cosine function is given by y = A cos ( B x ) , where:
∣ A ∣ represents the amplitude of the function.
B is related to the period (T) of the function by the formula T = B 2 π .
Identifying A and B In our given function, y = − 4 cos ( 3 2 π x ) , we can identify the values of A and B :
A = − 4
B = 3 2 π
Calculating Amplitude Now, let's calculate the amplitude. The amplitude is the absolute value of A :
A m pl i t u d e = ∣ A ∣ = ∣ − 4∣ = 4
Calculating Period Next, we'll calculate the period using the formula T = B 2 π :
T = 3 2 π 2 π = 2 π ⋅ 2 π 3 = 3
Final Answer Therefore, the amplitude of the function is 4, and the period is 3.
Examples
Understanding the period and amplitude of trigonometric functions is crucial in many real-world applications. For example, in electrical engineering, alternating current (AC) can be modeled using sinusoidal functions. The amplitude represents the maximum voltage or current, while the period represents the time it takes for one complete cycle. Similarly, in acoustics, sound waves can be modeled using trigonometric functions, where the amplitude corresponds to the loudness of the sound and the period corresponds to the frequency or pitch. By analyzing the period and amplitude, engineers and scientists can design and optimize systems involving oscillations and waves.
The amplitude of the function y = − 4 cos ( 3 2 π x ) is 4, and the period is 3. Therefore, we conclude that the amplitude is 4 and the period is 3. The final answer is: Amplitude = 4 , Period = 3 .
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