How to Write the Equation of a Sine Graph

How to Write the Equation of a Sine Graph

The general equation of a sine graph is \(y = a sin(b(x – h)) + k\), where \(a\) is the amplitude, \(b\) is the period, \(h\) is the horizontal shift, and \(k\) is the vertical shift.

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Step-by-step to write the equation of a sine graph

To find out how to write the equation of a sine graph, follow the step-by-step guide below:

  1. Amplitude: The amplitude is the distance from the maximum or minimum value of the function to the midline (average of the maximum and minimum values). It is represented by the variable \(“a”\) in the equation.
  2. Period: The period is the distance between consecutive maximum or minimum values of the function. The period of the sine function is \(2π\). It is represented by the variable \(“b”\) in the equation.
  3. Horizontal Shift: The horizontal shift is the movement of the graph horizontally. It is represented by the variable \(“h”\) in the equation.
  4. Vertical Shift: The vertical shift is the movement of the graph vertically. It is represented by the variable \(“k”\) in the equation.

For example, if you want to write the equation of a sine graph with an amplitude of \(2\), a period of \(4π\), a horizontal shift of \(3\) units to the right and a vertical shift of \(1\) unit up, the equation would be \(y = 2 sin(\frac{(x – 3)}{(4π)} ) + 1\)

You can also use a graphing calculator or a software like Desmos to help you graph the function and find the equation.

It’s important to note that sine and cosine functions have an amplitude between \(-1\) and \(1\) and a period of \(2×pi\), and that the domain of these functions is all the real numbers.

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