Problem 26 - Algebra 2

Question

Identify the period, range, and amplitude of each function. \(y=3 \cos \left(-\frac{\theta}{3}\right)\)

Step-by-Step Solution

Answer

The amplitude of the function is 3, the period of the function is \(6\pi\), and the range of the function is between -3 and 3.

Step 1: Identify the Amplitude

The amplitude of any trigonometric function \(f(x) = A \cos (Fx)\) or \(f(x) = A \sin (Fx)\) is given by |A|. In our case, A = 3. The amplitude is the absolute value of 3, which is 3.

Step 2: Identify the Period

The period of any trigonometric function can be computed as \(T= \frac{2\pi}{|F|}\). Our function is \(y=3 \cos \left(-\frac{\theta}{3}\right)\) so F = \(-\frac{1}{3}\). Substitute |F| = \(\frac{1}{3}\) in the formula, and it gives \( T= \frac{2\pi}{\frac{1}{3}} = 6\pi\). This is the period of the function.

Step 3: Identify the Range

The range of a cosine function is between -1 and 1. The range of \(y=A \cos (x)\) is between -A and 1A. Therefore, for our function with an amplitude of 3, the range is between -3 and 3.

Key Concepts

Amplitude

The amplitude is a crucial concept when discussing trigonometric functions like sine and cosine. In simple terms, amplitude refers to the height of the wave from its middle point (equilibrium position) to its peak. For any trigonometric function, like cosine or sine, the amplitude is denoted by the coefficient in front of these functions in their standard form.

• For a cosine function, such as \(y = A \cos(Fx)\) or for a sine function, the amplitude is represented by \(|A|\).
• Amplitude tells us how far up and down a function can move from its center line.
• In the example provided, the function \(y = 3 \cos\left(-\frac{\theta}{3}\right)\), the amplitude is 3.

This is because the coefficient next to the cosine function is \(A = 3\). The amplitude is always taken as a positive value, representing the maximum deviation from zero. If you picture this as a wave, it means the top of the wave is 3 units above and the bottom is 3 units below the midline.

Period of a Function

The period of a function refers to the amount of horizontal distance (in terms of input values) needed for the function to complete one full cycle of its pattern. This is a common concept in periodic functions like sine and cosine, which inherently repeat their shape after a certain interval.

• To determine the period, we use the formula \(T = \frac{2\pi}{|F|}\) for both sine and cosine functions.
• Here, \(F\) represents the frequency of the function, or how many cycles occur over a standard interval.

Looking at the function \(y = 3 \cos\left(-\frac{\theta}{3}\right)\):

• The frequency \(F\) is \(-\frac{1}{3}\).
• Thus, the absolute value \(|F|\) is \(\frac{1}{3}\).
• Plugging this into the period formula, \(T = \frac{2\pi}{\frac{1}{3}} = 6\pi\).

This 6\(\pi\) indicates the function takes 6\(\pi\) units along the theta-axis to repeat its cycle, forming its complete wave pattern.

Range of a Function

The range of a function provides us with the set of all possible output values (y-values) that the function can produce. For the cosine function, the default range is between -1 and 1. This range changes based on the function's amplitude.

• In an equation like \(y = A \cos(x)\), the range is scaled by the amplitude \(A\).
• The cosine will oscillate between -A and A due to vertical stretching or compressing caused by the amplitude.

For the function \(y = 3 \cos\left(-\frac{\theta}{3}\right)\):

• The amplitude is 3, so the range becomes \([-3, 3]\).
• This means for every value of \(\theta\), the function output will lie between -3 and 3.

Understanding the range is vital because it tells us how high or low the function can reach, indicating the extent of its variations on the graph.