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When learning about the amplitude of a sine function, imagine riding a Ferris wheel and how high it takes you above and below the central axis; that's akin to the amplitude in the sine world.

For the function \( y = A \sin(Bx) \), the amplitude is represented by the coefficient \( A \). It’s the absolute value because amplitude is always a positive measurement since it indicates how far the wave rises above and falls below its midline. In other terms, amplitude is the vertical stretch of the sine wave. The greater the amplitude, the higher and lower the wave peaks and troughs.

In the given exercise \( y = \frac{1}{4} \sin(2 \pi x) \), the amplitude is \( |\frac{1}{4}| \), resulting in an amplitude of \( 0.25 \). This means the wave only reaches a quarter above and below its central axis compared to a standard sine wave with an amplitude of 1.

Now let's tap into the rhythm of the sine wave, where frequency plays its part like the beats in a song. Frequency determines how often the sine wave cycles repeat over a unit interval on the horizontal axis.

The frequency in a sine function \( y = A \sin(Bx) \) is indicated by the coefficient \( B \) inside the function. In simple language, it’s the number of cycles the wave completes within a fixed length. The larger the frequency, the more waves can be fitted into the same space, making the wave appear more rapidly oscillating.

From the problem \( y = \frac{1}{4} \sin(2 \pi x) \), the frequency, denoted by \( B \), is the coefficient before \( x \), which is \( 2\pi \). This explains that the function has a higher frequency compared to the basic sine function where \( B = 1 \), resulting in a quicker cycle through the waves.

After noticing the rising and falling motion of amplitude and tapping to the frequency beat, it’s time to find out how long it takes for the Ferris wheel of sine to make one full turn—the period.

In mathematics, the period of a sine function \( y = A \sin(Bx) \) is calculated by \( T = \frac{2\pi}{|B|} \), where \( T \) represents the period and \( B \) is the frequency. This period is essentially the horizontal length required for the function to start repeating its values. It embodies the time taken for one complete cycle of the wave.

The original exercise gave us the function \( y = \frac{1}{4} \sin(2 \pi x) \) and asked to find the period. By applying the period formula, the period \( T \) is \( T = \frac{2\pi}{|2\pi|} = 1 \). This outcome implies that the sine wave will repeat its pattern every single unit along the x-axis, showcasing a brief interval between repetitions compared to a standard sine wave with a period of \( 2\pi \).