Equation Of Sine Function Amplitude 1/4 Period 4π Shift 4π

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In this article, we will delve into the process of determining the equation of a sine function based on specific parameters. These parameters include the amplitude, which dictates the maximum displacement from the sinusoidal axis; the period, which governs the length of one complete cycle; and the horizontal shift, also known as the phase shift, which determines the displacement of the function along the x-axis. Specifically, we will focus on a positive sine function, characterized by its initial upward direction from the sinusoidal axis. We'll break down each component and how it contributes to the final equation, ensuring a clear understanding of the transformations applied to the basic sine wave.

Understanding the General Form of a Sine Function

The general form of a sine function provides a framework for understanding how different parameters affect the graph's shape and position. It's expressed as:

y = A sin(B(x - C)) + D

Where:

  • A represents the amplitude, the vertical distance from the sinusoidal axis to the maximum or minimum point of the function. It dictates how tall the wave is.
  • B is related to the period (P) by the formula P = 2π / |B|. The period is the length of one complete cycle of the sine wave.
  • C represents the horizontal shift or phase shift. It determines how much the graph is shifted to the left or right. A positive C indicates a shift to the right, and a negative C indicates a shift to the left.
  • D represents the vertical shift, which moves the entire graph up or down. It's not directly relevant in this problem, but it's a crucial parameter for general sine functions.

This general form is the key to deciphering the equation of any sine function when given its characteristics. By carefully identifying each parameter, we can construct the equation that precisely represents the desired sinusoidal behavior. For the problem at hand, we'll focus on determining A, B, and C based on the given amplitude, period, and horizontal shift, respectively.

Determining the Amplitude (A)

The amplitude of a sine function is a crucial parameter that determines the vertical stretch of the wave. It represents the maximum displacement of the function from its sinusoidal axis, which is the horizontal line about which the sine wave oscillates. In simpler terms, it's the height of the wave from its center line. The amplitude is always a positive value, as it represents a distance. Given the problem statement, we are told that the amplitude of our sine function is $ rac{1}{4}$. This means that the sine wave will oscillate between $ rac{1}{4}$ units above and $ rac{1}{4}$ units below its sinusoidal axis. The sinusoidal axis is typically the x-axis (y = 0) unless there is a vertical shift (D in the general equation). Since no vertical shift is specified in the problem, we can assume the sinusoidal axis is the x-axis. The amplitude directly corresponds to the value of 'A' in the general form of the sine function, y = A sin(B(x - C)) + D. Therefore, in our case, A = $ rac{1}{4}$. This value will be the coefficient multiplying the sine function, directly scaling its vertical size. Understanding and correctly identifying the amplitude is essential as it sets the boundaries within which the sine wave will oscillate, influencing the overall visual representation and behavior of the function. In practical applications, the amplitude might represent the intensity of a sound wave, the voltage of an alternating current, or the height of a water wave. Therefore, accurately determining the amplitude is not only crucial for mathematical representations but also for understanding real-world phenomena modeled by sine functions.

Calculating the Value of B from the Period

The period of a sine function is another fundamental characteristic, defining the length of one complete cycle of the wave. It's the horizontal distance over which the function repeats its pattern. In the context of our problem, the period is given as $4 \pi$. This means that the sine wave will complete one full oscillation over an interval of $4 \pi$ units along the x-axis. The period is directly related to the coefficient 'B' in the general form of the sine function, y = A sin(B(x - C)) + D, through the formula: Period (P) = 2$ \pi$ / |B|. To determine the value of B, we need to rearrange this formula and substitute the given period. We have P = $4 \pi$, so: $4 \pi$ = 2$ \pi$ / |B|. Multiplying both sides by |B| and then dividing by $4 \pi$ gives us |B| = 2$ \pi$ / $4 \pi$ = $ rac{1}{2}$. Since we are looking for a positive sine function, we can take B = $ rac{1}{2}$. If B were negative, it would essentially reflect the sine function across the y-axis, but the period remains the same. Understanding this relationship between the period and B is crucial because it allows us to manipulate the horizontal compression or stretching of the sine wave. A smaller value of B stretches the wave horizontally, resulting in a longer period, while a larger value of B compresses the wave, resulting in a shorter period. In practical applications, the period might represent the time it takes for a pendulum to complete one swing, the wavelength of a light wave, or the duration of a musical note. Therefore, calculating B from the period is essential for accurately modeling these periodic phenomena.

Incorporating the Horizontal Shift (C)

The horizontal shift, also known as the phase shift, determines how much the sine function is translated to the left or right along the x-axis. It is represented by the parameter 'C' in the general form of the sine function, y = A sin(B(x - C)) + D. A positive value of C indicates a shift to the right, while a negative value of C indicates a shift to the left. In our problem, the horizontal shift is given as $4 \pi$ to the right. This directly translates to C = $4 \pi$ in the equation. It's important to note the (x - C) term in the general form, which means that a positive C will shift the function to the right, and a negative C will shift it to the left – the opposite of what might initially seem intuitive. The horizontal shift affects the starting point of the sine wave's cycle. For a standard sine function (y = sin(x)), the cycle begins at x = 0. However, with a horizontal shift of $4 \pi$ to the right, the cycle now begins at x = $4 \pi$. This shift does not change the shape or size of the wave; it merely repositions it along the x-axis. Understanding the horizontal shift is critical for accurately modeling periodic phenomena that begin at different points in time or space. For example, in electrical engineering, the phase shift between two alternating currents can be crucial for circuit performance. In acoustics, the phase difference between two sound waves can affect how they interfere with each other. Therefore, correctly incorporating the horizontal shift into the equation of a sine function is essential for representing real-world situations accurately.

Constructing the Final Equation

Now that we have determined the values of A, B, and C, we can construct the final equation of the sine function. We found that the amplitude (A) is $ rac{1}{4}$, the value of B is $ rac{1}{2}$, and the horizontal shift (C) is $4 \pi$. Substituting these values into the general form of the sine function, y = A sin(B(x - C)) + D, and noting that there is no vertical shift specified (so D = 0), we get:

y = $rac{1}{4}$ sin($rac{1}{2}$(x - $4
\pi$))

This equation represents a positive sine function with the specified characteristics. It has an amplitude of $ rac{1}{4}$, meaning it oscillates between -$ rac{1}{4}$ and +$ rac{1}{4}$. Its period is $4 \pi$, indicating that it completes one full cycle over an interval of $4 \pi$ units along the x-axis. And it has a horizontal shift of $4 \pi$ to the right, meaning its cycle starts $4 \pi$ units to the right of the standard sine function. This final equation encapsulates all the given parameters and accurately describes the desired sine wave. It's a powerful representation that can be used for graphing, analyzing, and predicting the behavior of the function. Understanding how to construct such equations from given parameters is fundamental in various fields, including physics, engineering, and signal processing, where sinusoidal functions are used to model a wide range of phenomena.

Summary

In conclusion, finding the equation of a sine function involves carefully considering the given parameters and applying them to the general form of the equation. The amplitude dictates the vertical stretch, the period determines the horizontal compression or expansion, and the horizontal shift repositions the function along the x-axis. By systematically determining each parameter – A, B, and C – we can construct an equation that accurately represents the desired sine wave. This process not only reinforces our understanding of sine functions but also provides a foundation for modeling and analyzing various periodic phenomena in the real world. The ability to translate descriptive parameters into a concise mathematical equation is a powerful skill in many scientific and engineering disciplines.