Problem 25 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The amplitude of the function \( y=\frac{1}{3} \cos \frac{\theta}{2} \) is \( \frac{1}{3} \), the period is \( 4\pi \), and the range is \( -\frac{1}{3} \leq y \leq \frac{1}{3} \).

Step 1: Identify Amplitude

Firstly, identify the amplitude of the function. The amplitude is the absolute value of the coefficient of cosine function, which is \( \frac{1}{3} \) here. So the amplitude is \( \frac{1}{3} \).

Step 2: Identify Period

Next, find the period of the function. The absolute value of the coefficient of \( \theta \) inside the cosine function is \( \frac{1}{2} \). The formula for the period of cosine function is \( \frac{2\pi}{|B|} \). Thus, the period for the provided function is \( \frac{2\pi}{|\frac{1}{2}|} = 4\pi \).

Step 3: Identify Range

Lastly, identify the range of the function. For the given function, there's no vertical shift. Thus, considering the amplitude (which is \( \frac{1}{3} \)), the range would be \( -\frac{1}{3} \leq y \leq \frac{1}{3} \).

Key Concepts

Amplitude

In trigonometry, the amplitude of a function is an essential feature to understand how the graph behaves. For the cosine function, the amplitude refers to the maximum height of the wave from its central axis.
In the function given, \(y=\frac{1}{3} \cos \frac{\theta}{2}\), the amplitude is determined by the coefficient in front of the cosine term. Here, the amplitude is \(\left|\frac{1}{3}\right|\), which is \(\frac{1}{3}\).

• The amplitude tells us how "tall" the wave is.
• A greater amplitude means a taller wave, while a smaller amplitude means a shorter wave.

Understanding amplitude helps visualize the function's behavior on a graph. It indicates how much the function values oscillate above and below the central line.

Period

The period of a trigonometric function is the length of one complete cycle on the graph.
The standard period of the cosine function is \(2\pi\), but multiplying \(\theta\) by a factor alters the period.
In the equation \(y=\frac{1}{3} \cos \frac{\theta}{2}\), we find this factor inside the cosine function. The coefficient beside \(\theta\) is \(\frac{1}{2}\).
To find the period, use the formula:
\[ \text{Period} = \frac{2\pi}{|B|} \]
Substituting \(B = \frac{1}{2}\), the new period becomes:
\[ \frac{2\pi}{\frac{1}{2}} = 4\pi \]
This means it takes \(4\pi\) units along the \(\theta\)-axis for the cosine wave to complete a full cycle.

Range

The range of a trigonometric function indicates the span of y-values it can take.
For a cosine function without a vertical shift, the range is influenced by its amplitude.
Consider the function \(y=\frac{1}{3} \cos \frac{\theta}{2}\). Here, the amplitude is \(\frac{1}{3}\), and there's no vertical shift, so the wave oscillates equally above and below the horizontal axis.
Using the amplitude, the standard range for cosine becomes:
\[-\frac{1}{3} \leq y \leq \frac{1}{3}\]
Key points to note:

• The range is confined between the negative and positive amplitude values.
• No vertical shifts means using just the amplitude to determine the range.

This understanding is crucial to accurately graph the function and predict its behavior.

Cosine Function

The cosine function is one of the basic trigonometric functions, crucial for modeling periodic phenomena.
With its regular wave pattern, the cosine function is defined generally as \(y = A \cos(B\theta + C) + D\).
The parameters influence the wave:

• \(A\) controls the amplitude.
• \(B\) adjusts the period.
• \(C\) shifts the phase.
• \(D\) causes a vertical shift.

In our specific case, \(y=\frac{1}{3} \cos \frac{\theta}{2}\), we observe:

• Amplitudes of \(\frac{1}{3}\) indicating how extreme the values get from the center.
• A period of \(4\pi\) showing how the function stretches horizontally.

The cosine function's properties make it invaluable for analyzing cycles, oscillations, and wave patterns in real-life applications.