Problem 28 - Algebra 2

Question

Identify the period, range, and amplitude of each function. \(y=16 \cos \frac{3 \pi}{2} t\)

Step-by-Step Solution

Answer

The amplitude of the function \(y = 16 \cos \frac{3 \pi}{2} t\) is 16, the period is \(\frac{4}{3}\), and the range is \([-16, 16]\).

Step 1: Find the Amplitude

The amplitude is the absolute value of the coefficient of the cosine function. For given function \(y=16 \cos \frac{3\pi}{2}t\), the amplitude is \(|16|\), which is 16.

Step 2: Find the Period

The period of the function is calculated using \(2\pi / |b|\), where b is the coefficient of t. In \(y=16 \cos \frac{3 \pi}{2} t\), b is \(3 \pi / 2\). Hence, period is \[2\pi / \left| \frac{3 \pi}{2} \right| = \frac{2\pi}{\frac{3 \pi}{2}} = \frac{4}{3}\].

Step 3: Find the Range

The range of a cosine function is easy as it always between \(-|A|\) and \(|A|\), where A is the amplitude. Here, the amplitude is 16 so the range of the function is from \(-16\) to \(16\). So, the range is [-16, 16].

Key Concepts

Amplitude

In trigonometric functions, the amplitude is a measure of the maximum distance a point on the graph reaches from the midline along the vertical axis. It essentially determines how "tall" the wave of the function appears. For the function given by \[y = 16 \cos \frac{3\pi}{2} t\], the amplitude is determined by the value that is multiplying the cosine function.

• To find the amplitude, simply take the absolute value of this multipliers.
  Here, the coefficient of the cosine function is 16. Thus, the amplitude is \(|16|\), which equals 16.
• Amplitude reveals the height of the peaks and the depth of the troughs relative to the central axis of the graph, usually represented as the x-axis.
• This information tells us that the graph of the cosine function will rise 16 units above and drop 16 units below the horizontal axis.

Period

The period of a trigonometric function is the length of the interval over which the function completes one full cycle of its periodic pattern. It reflects how "stretched" or "compressed" the wave of the function is.

• For the cosine function, the period is typically found by using the formula \[2\pi / |b|\], where \(b\) represents the coefficient of \(t\) in \[y = A \cos(bt)\].
• For our function, \[b = \frac{3\pi}{2}\] so the period is calculated as \[2\pi / \left| \frac{3\pi}{2} \right| = \frac{2\pi}{\frac{3\pi}{2}} = \frac{4}{3}\].
• This means the function completes one full wave cycle every \(\frac{4}{3}\) units along the horizontal axis.

Understanding the period is crucial for sketching the graph because it dictates the width of the cycles.

Range

The range of a trigonometric function describes the set of all possible output values (the y-values) that the function can produce. For a cosine function, the range is influenced by the amplitude and tells us the bounds of the graph on the y-axis.

• A basic unaltered cosine function oscillates between -1 and 1. When a coefficient, known as the amplitude, adjusts this oscillation, it scales the range accordingly.
• In our example, the amplitude is 16, hence influencing the range to be between \(-16\) and \(16\).
• Thus, for \[y = 16 \cos \left( \frac{3\pi}{2} t \right)\], the range expands from \(-16, 16\).This tells us that the highest point on the graph is \(16\) and the lowest point is \(-16\).

This information about the range helps to visualize the vertical extent of the graph and analyze the potential outputs for given input values.