# 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.

## 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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