How to Write the Equation of a Sine Graph

To write a sine equation from a graph, find the midline, amplitude, period, and a midline crossing. Use \(y=A\sin(B(x-h))+D\), where the amplitude is \(|A|\), the period is \(P=2\pi/|B|\), the horizontal shift is \(h\), and the midline is \(y=D\). The parameter \(B\) controls the period; it is not the period itself. These formulas use radians and assume \(A\ne0\), \(B\ne0\).

Related Topics

• How to Graph the Sine Function
• How to Graph the Cosine Function
• How to Graph Inverse of the Sine Function
• How to Graph Inverse of the Cosine Function

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: Read the maximum \(M\) and minimum \(m\). The amplitude is \((M-m)/2\), a nonnegative distance. In the equation it equals \(|A|\), not necessarily \(A\).
2. Period: Measure the horizontal distance \(P\) between consecutive peaks, consecutive troughs, or matching crossings in the same direction. Choose \(B=2\pi/P\). For an axis measured in degrees, use \(B=360^\circ/P\) instead.
3. Horizontal shift: For positive \(A\) and \(B\), choose \(h\) at a midline crossing where the graph rises. A downward crossing requires a negative \(A\) with this choice of positive \(B\). Equivalent equations may use a different crossing.
4. Vertical shift: Set \(D=(M+m)/2\), the height of the midline.

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For amplitude \(2\), period \(4\pi\), a right shift of \(3\), and an upward shift of \(1\), choose \(A=2\), \(B=2\pi/(4\pi)=1/2\), \(h=3\), and \(D=1\). A matching equation is \(y=2\sin\left(\frac{x-3}{2}\right)+1\). Check: its period is \(2\pi/(1/2)=4\pi\), its maximum is \(3\), and its minimum is \(-1\).

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

The parent functions \(y=\sin x\) and \(y=\cos x\) have amplitude \(1\), range \([-1,1]\), period \(2\pi\), and domain all real numbers. Amplitude is a distance; the interval from \(-1\) to \(1\) describes their range.

Worked example: find an equation from key points

Suppose a sinusoidal graph has maximum \(5\), minimum \(-1\), period \(8\), and crosses the midline upward at \(x=2\). Then \(D=(5+(-1))/2=2\), \(A=(5-(-1))/2=3\), \(B=2\pi/8=\pi/4\), and \(h=2\). One equation is \(y=3\sin\left(\frac{\pi}{4}(x-2)\right)+2\).

Check five equally spaced points in one cycle: \((2,2)\), \((4,5)\), \((6,2)\), \((8,-1)\), and \((10,2)\). Their horizontal spacing is \(P/4=2\). These points confirm the upward crossing, maximum, minimum, and return to the starting height and direction.

Check the equation before you finish

Substitute a peak, a trough, and a midline crossing from the graph. Confirm the maximum and minimum, one full repeat length, and whether the graph rises or falls at your chosen crossing. In \(A\sin(Bx-C)+D\), the horizontal shift is \(C/B\); factor out \(B\) before reading the shift.

For the underlying transformation formulas, see OpenStax: Graphs of the Sine and Cosine Functions. Use the related sine and cosine lessons above to compare equivalent representations.