Problem 16 - Precalculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Amplitude

The amplitude of a sine function, which is of the form \( y = a \sin(bx) \), is determined by the absolute value of the coefficient \( a \). In this case, the function is \( y = \frac{3}{2} \sin\left(\frac{2}{3} x\right) \), so the amplitude is \( \left| \frac{3}{2} \right| = \frac{3}{2} \).

Step 2: Determine the Period

The period of a sine function is found using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the sine function. For the given function, \( b = \frac{2}{3} \). Therefore, the period is \( \frac{2\pi}{\frac{2}{3}} = 3\pi \).

Key Concepts

Amplitude

Amplitude refers to the height of the wave produced by a trigonometric function like the sine. Specifically, it measures the distance the wave extends vertically from its central position to its peak or trough. For functions such as \( y = a \sin(bx) \), you determine the amplitude by calculating the absolute value of coefficient \( a \).

In our example, we have the sine function \( y = \frac{3}{2} \sin\left(\frac{2}{3} x\right) \). Here, the coefficient \( a \) is \( \frac{3}{2} \). Therefore, the amplitude can be found by:

• Taking the absolute value of \( \frac{3}{2} \)
• Computing \( \left| \frac{3}{2} \right| = \frac{3}{2} \)

The amplitude tells us how tall each "hill" and "valley" of the sine wave rise above or fall below the center line of the graph.

Period

The period of a sine function tells us how long it takes for the wave to complete one full cycle. This is the horizontal length of one complete wave. For sine waves, calculating the period involves the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient that modifies \( x \) in the sine function.

In our case, we are working with the function \( y = \frac{3}{2} \sin\left(\frac{2}{3} x\right) \). Here, \( b \) is \( \frac{2}{3} \). To find the period:

• Apply the formula: \( \frac{2\pi}{b} \)
• Substitute \( b = \frac{2}{3} \) into the equation, obtaining \( \frac{2\pi}{\frac{2}{3}} = 3\pi \)

Thus, the period is \( 3\pi \), meaning the wave repeats itself every \( 3\pi \) units along the x-axis.

Sine Function

The sine function is one of the main trigonometric functions and plays a vital role in describing periodic phenomena. Its general form is \( y = a \sin(bx) + c \), where:

• \( a \) affects the amplitude (height of the wave)
• \( b \) influences the period (length of one complete cycle)
• \( c \) determines vertical shifts, moving the graph up or down

In essence, the sine function outputs values based on the input angle \( x \), rendering it periodic, which means it repeats its pattern at regular intervals.

Sine waves are characterized by their smooth oscillating pattern. They begin at the origin \( (0,0) \), rise to a peak, dip to a trough, and then return to the axis, completing one full cycle over its period. It's this repeating quality that makes sine functions so essential for modeling waves, whether they be sound, light, or even tides.