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Chapter 8: Problem 1

find the period and amplitude. $$ y=2 \sin 2 x $$

Short Answer

Expert verified
The amplitude of the function \(y=2 \sin 2x\) is 2 and the period is \(\pi\).

Step by step solution

01

Finding the amplitude

Look at the coefficient attached to the \(\sin\) term of given function, which is 2. Thus, the amplitude of the function \(y=2\sin2x\) is 2.
02

Finding the period

Calculate the period of the function, using the formula for the period of a sinusoidal function which is \(\frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\). Therefore, the period of the function is \(\frac{2\pi}{|2|} = \pi\)

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Key Concepts

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

Amplitude
In the context of sinusoidal functions, amplitude refers to the peak value of the wave, or how "tall" the wave appears. For the function \(y = 2 \sin 2x\), the amplitude is given by the coefficient in front of the sine function. Here, the coefficient is 2.
This means that the graph of the function will reach a maximum height of 2 above the horizontal axis and a minimum depth of 2 below the horizontal axis.
Amplitude can be thought of as the distance from the middle of the wave (often called the midline or equilibrium position) to its peak.
  • The amplitude is always a positive number.
  • It determines the wave's intensity or loudness in sound waves.
  • In graphs, it visually represents how far the wave travels vertically from its center.
Period of Sinusoidal Functions
The period of a sinusoidal function is the distance along the x-axis required for the function to complete one full cycle.
For the sine function \(y = \sin x\), the default period is \(2\pi\), meaning it takes \(2\pi\) units to complete one cycle. In the function \(y = 2 \sin 2x\), we modify the period by introducing the coefficient 2 alongside \(x\).
To determine the period of the function \(y = 2 \sin 2x\), use the formula \(\frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\). In this function, \(B\) is equal to 2. Thus, the period becomes \(\frac{2\pi}{2} = \pi\).
  • The period determines the horizontal stretch or compression of the wave.
  • Shorter periods result in more cycles over a given interval.
  • In practice, the period might denote time for waves, such as sound frequencies or light waves.
Sinusoidal Function
Sinusoidal functions, such as \(y = a \sin(bx + c) + d\) or \(y = a \cos(bx + c) + d\), are fundamental in modeling periodic behaviors. These functions are called sinusoidal because they resemble the sine or cosine wave patterns.
Each component in the formula affects the graph:
  • \(a\) changes the amplitude, making the wave taller or shorter.
  • \(b\) adjusts the period, compressing the wave horizontally when larger.
  • \(c\) moves the graph left or right, shifting the phase.
  • \(d\) translates the graph up or down, modifying the midline.
A sinusoidal function is frequently used to model everything from sound waves to tides and seasons due to its repetitive, smooth oscillatory nature. These functions provide insights into cyclic phenomena and are vital in fields such as engineering, physics, and environmental science.

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Most popular questions from this chapter

solve the equation for \(\theta\) \((0 \leq \theta \leq 2 \pi) .\) For some of the equations you should use the trigonometric identities listed in this section. Use the trace feature of a graphing utility to verify your results. $$ \sin 2 \theta-\cos \theta=0 $$

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