Problem 11 - Precalculus

Question

In Exercises \(11-20,\) state the amplitude and period of each function. $$y=\frac{3}{2} \cos (3 x)$$

Step-by-Step Solution

Answer

Amplitude is \(\frac{3}{2}\); Period is \(\frac{2\pi}{3}\).

Step 1: Identify the Standard Form

The standard form for a cosine function is given by \(y = a \cos(bx)\). In this case, the function is \(y = \frac{3}{2} \cos(3x)\). By comparing, we identify \(a = \frac{3}{2}\) and \(b = 3\).

Step 2: Determine the Amplitude

The amplitude of a trigonometric function like cosine is the absolute value of the coefficient \(a\). Here, \(a = \frac{3}{2}\), so the amplitude is \(\left| \frac{3}{2} \right| = \frac{3}{2}\).

Step 3: Determine the Period

The period of a cosine function is calculated as \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) in the function. In this case, \(b = 3\), so the period is \(\frac{2\pi}{3}\).

Key Concepts

Amplitude

The amplitude of a trigonometric function is an essential characteristic that reflects how high and low the function graph goes. For the cosine function, represented as \( y = a \cos(bx) \), the amplitude is determined by the coefficient \( a \) in front of the cosine function.

• Amplitude is always a positive number, even if the coefficient \( a \) is negative.
• It specifies the maximum vertical distance of the cosine curve from the horizontal axis (the x-axis).
• The amplitude affects how 'tall' or 'short' the waves of the graph appear.

In our exercise, the function \( y = \frac{3}{2} \cos (3x) \) has an amplitude equal to \( \left| \frac{3}{2} \right| = \frac{3}{2} \). This means the waves of the cosine graph will reach a maximum height of 1.5 units and a minimum depth of -1.5 units from the center line.

Period

The period of a trigonometric function is another critical attribute that dictates the horizontal length of one complete cycle of the graph. For the cosine function, in the standard form, \( y = a \cos(bx) \), the period is calculated using \( \frac{2\pi}{b} \).

• The period represents how much of the x-axis is covered until the function begins to repeat itself.
• It defines the distance between two consecutive maximum points (peaks) or two consecutive minimum points (troughs) on the graph.
• In a basic cosine function, without any transformations, the period is \( 2\pi \).

For our function \( y = \frac{3}{2} \cos(3x) \), the coefficient \( b = 3 \), hence the period is \( \frac{2\pi}{3} \). This means that the cosine waves complete a full cycle every \( \frac{2\pi}{3} \) units along the x-axis.

Cosine Function

The cosine function, a fundamental element of trigonometry, showcases periodic behavior and is widely used to model wave patterns and oscillations. A standard cosine function is expressed in the form \( y = a \cos(bx) \), featuring specific characteristics based on amplitude and period.

• The cosine graph is a smooth, wave-like curve that oscillates above and below the x-axis.
• When graphing, it starts from its maximum value at \( x = 0 \) if there are no horizontal shifts.
• It has key points at the maximum (amplitude), minimum (negative amplitude), and when it crosses the x-axis (zero).

In our exercise, the function \( y = \frac{3}{2} \cos (3x) \) involves a vertical scaling (amplitude \( \frac{3}{2} \)) and a horizontal compression (period \( \frac{2\pi}{3} \)), which means the wave pattern is taller and repeats itself more frequently than the basic cosine function. Understanding these transformations helps in graphing and analyzing real-world phenomena like sound and light waves.