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Chapter 7: Problem 17

Determine the amplitude and period of each function without graphing. $$ y=6 \sin (\pi x) $$

Short Answer

Expert verified
Amplitude: 6, Period: 2

Step by step solution

01

Identify the general form of the sine function

A sine function is generally written as \[y = A \sin(Bx + C) + D\]where:\A = \text{Amplitude},\B = \text{Frequency factor}.\
02

Determine the amplitude

The amplitude is the coefficient in front of the sine function, which is the absolute value of A.From the given equation \[y = 6 \sin ( \pi x )\], the amplitude |A| is |6|.Therefore, the amplitude is 6.
03

Identify the frequency factor

In the given equation \[y = 6 \sin ( \pi x )\], the frequency factor B is \( \pi \).
04

Calculate the period

The period of the sine function is calculated as \text{Period} = \frac{2\pi}{B}nFrom the previous step, B = \( \pi \).Thus, \text{Period} = \frac{2\pi}{\pi} = 2.Therefore, the period is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sine function
The sine function is a fundamental concept in trigonometry. It models periodic phenomena like sound waves and tides. The general form is \( y = A \sin(Bx + C) + D \) where: - \( A \) is the amplitude, - \( B \) affects the period and frequency, - \( C \) shifts the function horizontally, and - \( D \) shifts it vertically. Understanding each parameter helps in graphing and analyzing sine waves.
amplitude
The amplitude is the height from the center line to the peak (or trough) of the sine wave. It is always positive since it's the absolute value of the coefficient in front of the sine function. In the equation \( y = 6 \sin(\pi x) \), the amplitude is \( |6| \). Thus, the amplitude is 6. It represents the maximum displacement from the central axis and determines how 鈥渢all鈥� or 鈥渟hort鈥� the wave appears.
period
The period of a sine wave is the length of one complete cycle. It is calculated using the formula: \( \text{Period} = \frac{2\pi}{B} \) For \( y = 6 \sin(\pi x) \), the frequency factor \( B \) is \( \pi \). Therefore, the period is \( \text{Period} = \frac{2\pi}{\pi} = 2 \). This means the wave completes one full cycle every 2 units along the x-axis.
frequency factor
The frequency factor \( B \) in the function \( y = A \sin(Bx + C) + D \) affects the width of the cycles. Higher \( B \) values lead to shorter periods and more frequent cycles, while lower \( B \) values result in longer periods and fewer cycles. In \( y = 6 \sin(\pi x) \), the frequency factor \( B \) is \( \pi \). This tells us how often the sine function cycles occur within a given interval on the x-axis.

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