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Chapter 7: Problem 442

For the following exercises, find the amplitude, frequency, and period of the given equations. $$ y=-2 \sin (16 x \pi) $$

Short Answer

Expert verified
Amplitude: 2, Frequency: 8, Period: \(\frac{1}{8}\)

Step by step solution

01

Identify the Function

The given equation is:\( y = -2 \sin(16 x \pi) \).This is a sine function of the form \( y = a \sin(bx) \), where \( a = -2 \) and \( b = 16 \pi \).
02

Find the Amplitude

The amplitude of a sine function \( y = a \sin(bx) \) is the absolute value of \( a \).Here, \( a = -2 \).So, the amplitude is \( |a| = 2 \).
03

Find the Frequency

The frequency of the function is given by the term \( b \) in \( \sin(bx) \).So in this case, \( b = 16\pi \), which means the frequency is \( \frac{b}{2\pi} = \frac{16\pi}{2\pi} = 8 \).
04

Find the Period

The period of a sine function is given by \( \frac{2\pi}{b} \).Since \( b = 16\pi \), the period is \( \frac{2\pi}{16\pi} = \frac{1}{8} \).

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Key Concepts

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

Amplitude
The amplitude of a trigonometric function is one of its most defining characteristics. It represents the maximum distance a function reaches from its equilibrium or center line.
For the sine function, expressed as \( y = a \sin(bx) \), the amplitude is the absolute value of \( a \). This indicates how far the peaks and troughs of the wave go above and below the centerline.
  • For the equation \( y = -2 \sin(16x\pi) \), the coefficient \( a \) is \(-2\).
  • The amplitude, therefore, is \( |a| = 2 \).
This means, no matter how the sine function is shifted vertically, its peaks will always be 2 units above and its lowest points 2 units below the central axis.
Frequency
Frequency in the context of trigonometric functions refers to how many complete cycles of the waveform occur within a unit of time.
It is a measure of how rapidly the function oscillates.For a sine function \( y = a \sin(bx) \), the frequency is determined by the coefficient \( b \), and is calculated as the number of wave cycles that happen in \( 2\pi \) radians:
  • Using the given equation \( y = -2 \sin(16x\pi) \), the coefficient \( b \) is \( 16\pi \).
  • The frequency is calculated as \( \frac{b}{2\pi} = \frac{16\pi}{2\pi} = 8 \).
Thus, the sine function completes 8 cycles per interval of \( 2\pi \).
Higher frequency means the function oscillates more rapidly, making the wave appear more compressed horizontally.
Period
The period of a trigonometric function signifies the length of one complete cycle on the horizontal axis.
For a sine wave, it is crucial as it tells us how long it takes to repeat its pattern.In the general form \( y = a \sin(bx) \), the period is calculated by taking \( \frac{2\pi}{b} \):
  • The given function \( y = -2 \sin(16x\pi) \) has a \( b \) value of \( 16\pi \).
  • Thus, its period is \( \frac{2\pi}{16\pi} = \frac{1}{8} \).
This means the function repeats its cycle every \( \frac{1}{8} \) units along the x-axis.
A shorter period indicates that the wave pattern repeats more frequently, showing more cycles in a given interval.

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Most popular questions from this chapter

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