Problem 27 - Algebra 2

Question

Identify the period, range, and amplitude of each function. \(y=-\frac{1}{2} \cos 3 \theta\)

Step-by-Step Solution

Answer

The amplitude of the function \(y=-\frac{1}{2} \cos 3 \theta\) is \(\frac{1}{2}\), the period is \(2\pi / 3\), and the range is from \(-\frac{1}{2}\) to \(\frac{1}{2}\).

Step 1: Identify the Coefficient of Cos and Calculate Amplitude

In the given function, \(y=-\frac{1}{2} \cos 3 \theta\), the coefficient of cosine is \(-\frac{1}{2}\). The amplitude of the function is the absolute value of this coefficient, which is \(\frac{1}{2}\).

Step 2: Identify the Coefficient of Theta and Calculate Period

The coefficient of \(\theta\) in the function is 3. Therefore, the period of the cosine function is \(2\pi / 3\).

Step 3: Determine the Range

The range of the function is from negative amplitude to positive amplitude, i.e, from \(-\frac{1}{2}\) to \(\frac{1}{2}\).

Key Concepts

Amplitude

Amplitude is an essential characteristic of trigonometric functions, specifically sine and cosine. It measures how much the function deviates from its central axis or average value. Think of amplitude as the "height" of the wave peak or depth of the trough in a graph. It provides insight into the strength of the wave.

In the function \(y=-\frac{1}{2} \cos 3 \theta\), the amplitude is derived from the coefficient in front of the cosine, which is \(-\frac{1}{2}\). To find the amplitude:

• Take the absolute value of the coefficient: \(|-\frac{1}{2}|\)
• Resulting amplitude is \(\frac{1}{2}\)

It’s important to note that amplitude is always a positive number because it represents a measurement of distance. Regardless of whether the original coefficient is negative or positive, you always take its absolute value to determine amplitude.

Period

The period of a trigonometric function is the distance (or length) required for the function to complete one full cycle. For cosine functions, the standard period is typically \(2\pi\), which represents a full 360-degree rotation.

However, when there is a coefficient multiplying the variable (in this case, \(3\)), the period changes. To calculate the period of the function \(y = -\frac{1}{2} \cos 3\theta\):

• Use the formula \(\frac{2\pi}{|b|}\), where \(b\) is the coefficient of \(\theta\).
• Here, \(b\) is \(3\), so the period is \(\frac{2\pi}{3}\).

This adjustment occurs because the function \(3\theta\) oscillates faster than the typical cosine function, completing more cycles over the same interval, therefore resulting in a shorter period.

Range

The range of a trigonometric function reflects the set of possible output values (y-values) that the function can take. For sine and cosine functions, this is influenced by the amplitude because it determines the maximum and minimum values that the function can reach.

For the function \(y=-\frac{1}{2} \cos 3 \theta\), the amplitude of \(\frac{1}{2}\) directly impacts the range:

• Standard range for cosine is from \(-1\) to \(1\).
• Multiply this standard range by the amplitude \(\frac{1}{2}\): \([-\frac{1}{2}, \frac{1}{2}]\).
• The negative sign in the function does not affect the range; it only reflects it over the x-axis.

Thus, the range of the function \(y=-\frac{1}{2} \cos 3 \theta\) is from \(-\frac{1}{2}\) to \(\frac{1}{2}\), meaning the function only oscillates within this band of values.