Problem 41 - Algebra 2

Question

Which function has a period of 2\(\pi\) and an amplitude of 4\(?\) F. \(f(x)=2 \cos 4 \theta\) . \(f(x)=2 \cos \theta\) H. \(f(x)=4 \cos 2 \theta \quad\) J. \(f(x)=4 \cos \theta\)

Step-by-Step Solution

Answer

The function which has a period of \(2\pi\) and an amplitude of 4 is \(f(x) = 4 \cos x\) (option J).

Step 1: Reviewing Mathematical Definitions

Remind that the amplitude of sinusoidal functions like \(f(x) = a \cos(bx)\) or \(f(x) = a \sin(bx)\) is the absolute value of the coefficient \(a\), and the period is \(\frac{2\pi}{|b|}\).

Step 2: Analyzing Option F

In the function \(f(x)=2 \cos 4x\), the absolute value of the coefficient in front of \(\cos x\) (the cosine function) is 2, so the amplitude is 2, not 4. The period is \(\frac{2\pi}{4}=\frac{\pi}{2}\), not \(2\pi\). Therefore, option F does not meet the requirements.

Step 3: Analyzing Option G

In the function \(f(x)=2 \cos x\), the amplitude is 2 and the period is \(2\pi\). Only the period matches the required period, but not the amplitude. Therefore, option G does not meet the requirements.

Step 4: Analyzing Option H

In the function \(f(x) = 4 \cos 2x\), the amplitude is 4, which matches the required amplitude. The period is \(\frac{2\pi}{2}=\pi\), not \(2\pi\). Therefore, option H does not meet the requirements.

Step 5: Analyzing Option J

In the function \(f(x) = 4 \cos x\), the amplitude is 4 and the period is \(2\pi\), which matches both the required amplitude and period. Therefore, option J meets the requirements.

Key Concepts

Amplitude

The term **amplitude** is a key concept when discussing trigonometric functions, particularly sinusoidal functions like sine and cosine. Amplitude refers to the height of the wave from its central axis.
In simpler terms, it is the maximum distance the wave reaches from its equilibrium point. This is important as it tells you how "tall" or "short" the wave is.

• Mathematically, the amplitude of a function like \( f(x) = a \cos(bx) \) is represented by the absolute value of \( a \).
• So if \( a = 4 \), the amplitude is 4, which means the wave reaches 4 units above and below its central position.

Amplitude is always a positive value, as it represents a distance, and it plays a crucial role in determining the overall shape and scale of sinusoidal functions. By understanding amplitude, we better grasp the magnitude of oscillation present in the wave.

Period of a Function

The **period of a function** describes the length it takes for a function to complete one full cycle. It is most commonly discussed in the context of periodic functions, like the trigonometric functions sine and cosine. These functions repeat at regular intervals, and the period tells us when these repetitions occur.

• For standard trigonometric functions, the period is related to the coefficient \( b \) in functions of the form \( f(x) = a \cos(bx) \).
• The period is calculated using the formula \( \frac{2\pi}{|b|} \).
• This means, for a function \( f(x) = a \cos(bx) \) where \( b = 1 \), the period is \( 2\pi \).

Understanding the period is crucial for predicting the behavior of oscillating phenomena. It helps in determining how often the function returns to its starting point, which is valuable for create models and solving problems involving cycles.

Cosine Function

The **cosine function** is one of the fundamental trigonometric functions, known for its periodic and symmetrical properties. It helps in defining relationships in various fields such as physics, engineering, and even finance.
The cosine function is written as \( \, \cos(x) \) and is most commonly seen in models and analyses that involve waves or cyclical patterns.

• It is characterized by its periodic nature, repeating every \( 2\pi \) radians.
• The graph of the cosine function oscillates between -1 and 1, providing smooth wave-like curves that return to the same point.
• When expressed in the format \( f(x) = a \cos(bx) \), the cosine function is adjusted in amplitude and frequency by the coefficients \( a \) and \( b \).

The cosine function's importance lies in its ability to model and predict various periodic behaviors. Whether you are analyzing alternating current in electronics or predicting natural cycles, understanding how the cosine function behaves is essential.