Problem 15 - Precalculus

Question

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

Step-by-Step Solution

Answer

The amplitude is \( \frac{2}{3} \) and the period is \( \frac{4\pi}{3} \).

Step 1: Identify the General Form

The general form for a cosine function is \( y = a \cos(bx) \) where \( a \) determines the amplitude and \( b \) influences the period of the function. Here, the function given is \( y = \frac{2}{3}\cos\left(\frac{3}{2}x\right) \).

Step 2: Determine the Amplitude

In the general form \( y = a \cos(bx) \), the amplitude is the absolute value of \( a \). Therefore, for \( y = \frac{2}{3}\cos\left(\frac{3}{2}x\right) \), the amplitude is \( \left| \frac{2}{3} \right| = \frac{2}{3} \).

Step 3: Calculate the Period

The period of a cosine function is given by \( \frac{2\pi}{b} \). In the function \( y = \frac{2}{3}\cos\left(\frac{3}{2}x\right) \), \( b = \frac{3}{2} \). Thus, the period is calculated as \( \frac{2\pi}{\frac{3}{2}} = \frac{2\pi \times 2}{3} = \frac{4\pi}{3} \).

Key Concepts

Amplitude

When dealing with trigonometric functions like a cosine function, amplitude is a key characteristic that tells us how "tall" or "short" the wave of the function is. This measurement is taken from the mid-line of the wave to its highest or lowest point. In mathematical terms, the amplitude is defined as the absolute value of the coefficient in front of the cosine function.
For example, in the equation

• \( y = a \cos(bx) \)

"a" is the coefficient, and the amplitude is \( |a| \). In our specific function,

• \( y = \frac{2}{3} \cos\left(\frac{3}{2}x\right) \)

the amplitude is \( \left| \frac{2}{3} \right| = \frac{2}{3} \). Here, the amplitude of \( \frac{2}{3} \) indicates that the wave will be \( \frac{2}{3} \) units above and below the mid-line (generally the x-axis in simple graphing scenarios), creating a wave that is not as tall as a standard cosine wave which would have an amplitude of 1.

Period

The period of a cosine function tells us the length over which the wave repeats itself. In simpler words, it is the horizontal distance before the wave pattern starts again. For cosine functions, this is calculated using the formula:

• \( \text{Period} = \frac{2\pi}{b} \)

where "b" is the coefficient of the variable inside the cosine function. This is derived from the standard wave, which has a period of \( 2\pi \). For any cosine function, we need to adjust this standard period value by dividing by the coefficient to get the correct period.
In the given function

• \( y = \frac{2}{3} \cos\left(\frac{3}{2}x\right) \)

we identify \( b \) as \( \frac{3}{2} \). Applying the formula gives us \[ \text{Period} = \frac{2\pi}{\frac{3}{2}} = \frac{4\pi}{3} \].This means that the wave of the function completes one full cycle every \( \frac{4\pi}{3} \) units along the x-axis. It shows that the pattern of the wave is elongated compared to the standard cosine period.

Cosine Function

The cosine function is one of the fundamental trigonometric functions often used to describe periodic phenomena like sound waves or alternating electrical currents. It has a wave-like shape and always follows a predictable pattern. The standard form for a cosine function is

• \( y = a \cos(bx) \)

where:

• "a" affects the amplitude, or the height of the wave,
• "b" determines the frequency or number of cycles in a given interval, thus influencing the period.

A standard cosine function, \( \cos(x) \), has a peak of 1, trough of -1, with a period of \( 2\pi \), completing one full cycle in \( 2\pi \) intervals.
In modified cosine functions like

• \( y = \frac{2}{3} \cos\left(\frac{3}{2}x\right) \)

the pattern alters due to the coefficients. The amplitude of \( \frac{2}{3} \) narrows the wave vertically, while the inside term \( \frac{3}{2}x \) changes how often the pattern repeats, making the period \( \frac{4\pi}{3} \), and indicating a stretch along the x-axis. Understanding these parameters allows us to graph and predict the behavior of such functions in different applications.