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Chapter 4: Problem 9

In Exercises 5-18, find the period and amplitude. \(y\ =\ \frac{1}{2} sin\ \frac{\pi x}{3}\)

Short Answer

Expert verified
The amplitude of the function y=\(\frac{1}{2} sin \frac{\pi x}{3}\) is \(\frac{1}{2}\) and the period is 6.

Step by step solution

01

Identify the amplitude

Amplitude is given by the absolute value of the coefficient of the trigonometric function. In this case, it is the absolute value of \(\frac{1}{2}\). Therefore, the amplitude is \(\frac{1}{2}\).
02

Identify the coefficient of x in the argument of the sine function

The coefficient of 'x' in the argument of the sine function is \(\frac{\pi}{3}\). Call this value 'B'.
03

Calculate the period

The period of a sine function is given by \(\frac{2\pi}{B}\). Here, 'B' is \(\frac{\pi}{3}\). So, the period is \(\frac{2\pi}{\frac{\pi}{3}}\) = 6.

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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 like sine represents the maximum distance it reaches from its central axis, which is often the horizontal x-axis. Amplitude is important because it gives us a sense of how much a wave-like curve 'grows' from its mid-level to the peak.
To find the amplitude, simply look at the coefficient in front of the sine function. This coefficient directly influences how tall the waves of the function appear.
  • For instance, in the function given by the exercise, which is: \(y = \frac{1}{2} \sin \frac{\pi x}{3}\)
  • The coefficient in front of the sine function is \(\frac{1}{2}\).
  • Therefore, the amplitude is \(\left| \frac{1}{2} \right| = \frac{1}{2}\).
This means that the wave reaches up to \(\frac{1}{2}\) units above and below its central axis, providing a visual cap to its oscillations.
Period of a Function
The period of a function is the interval after which the function's shape restarts. In trigonometric functions like sine or cosine, the period dictates how frequently the wave repeats itself as you move along the x-axis.
Determining the period helps in graphing the function accurately.
To find the period of a sine function represented as \( y = a \sin(bx) \), you use the formula:
  • \( \text{Period} = \frac{2\pi}{B} \) , where \(B\) is the coefficient of \(x\).
  • In our function \(y = \frac{1}{2} \sin \frac{\pi x}{3}\), \(B = \frac{\pi}{3}\).
  • This gives us a period of \(\frac{2\pi}{\frac{\pi}{3}} = 6\).
Thus, the wave pattern of this sine function will restart every 6 units along the x-axis.
Sine Function
The sine function, often denoted as \( \sin(x) \), is one of the fundamental functions used in trigonometry. It describes the smooth and repetitive oscillations seen in waves and circles.
Key characteristics include:
  • It has a periodic nature, meaning its pattern repeats after a set interval called the period.
  • Its range lies between -1 and 1 for the basic sine function, resulting in a wavelike appearance.
In the modified sine function \( y = \frac{1}{2} \sin \frac{\pi x}{3} \):
  • The amplitude \(\frac{1}{2}\) adjusts how tall the waves are, so they reach only halfway of typical unit height.
  • The period \(6\) defines how long it takes for the wave to complete a single cycle and restart.
  • This gives a unique waviness to the sine function, with adjusted peaks and cycle lengths crafted by those coefficients.
Understanding these components allows us to predict the wave's behavior as it travels across the graph.

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