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.

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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.

Reading each value off a sine graph

If the graph crosses the midline going down at your chosen point, keep h at that point and make A negative. If the graph starts at a peak, a cosine equation is often easier: y = A cos(B(x – h)) + D with h at the peak.

Example: a sine graph that starts by going down

A graph has a maximum of 4 and a minimum of -2. It crosses the midline going down at x = 0, and the next downward crossing is at x = π.

1. Midline: D = (4 + (-2)) ÷ 2 = 1.
2. Amplitude: (4 – (-2)) ÷ 2 = 3.
3. Period: π – 0 = π, so B = 2π ÷ π = 2.
4. The graph goes down at x = 0, so use A = -3 with no shift.

The equation is y = -3 sin(2x) + 1. Check two points. At x = π/4, y = -3 sin(π/2) + 1 = -2, the minimum. At x = 3π/4, y = -3 sin(3π/2) + 1 = 4, the maximum. Both match. Another correct answer is y = 3 sin(2(x – π/2)) + 1, which uses the upward crossing at x = π/2.

Frequently Asked Questions

What is the equation of a sine graph?

A sine graph follows y = A sin(B(x − h)) + D. The amplitude is |A|, the period is 2π/|B|, h is the horizontal shift and y = D is the midline. For example, y = 4 sin(2(x − 1)) − 3 has amplitude 4, period π, a shift of 1 to the right and midline y = −3.

How do you find B in a sine graph?

Measure the period P, the horizontal distance between two neighboring peaks, then use B = 2π/P. If a graph repeats every 4π, B = 2π/4π = 1/2. If it repeats every π, B = 2. When the x-axis is in degrees, use B = 360°/P instead. Remember that B is not the period itself.

How do you write a sine function from a graph?

Read the max and min, then the period and an upward midline crossing. Suppose the max is 5 and the min is −1: D = (5 + (−1))/2 = 2 and A = (5 − (−1))/2 = 3. If the period is 8, B = 2π/8 = π/4. With an upward crossing at x = 1, the equation is y = 3 sin(π/4 (x − 1)) + 2.

How do you write a sinusoidal function?

A sinusoidal function can use sine or cosine: y = A sin(B(x − h)) + D or y = A cos(B(x − h)) + D. Use sine when you can spot a point where the graph crosses its midline going up, and cosine when you can spot a maximum. Both forms give the same curve with a different horizontal shift.

What is the sine wave graph equation in physics?

In physics a sine wave is often written y = A sin(2πft + φ), where A is the amplitude, f is the frequency in cycles per second and φ is the phase shift. Here B = 2πf, so the period is 1/f. A wave with amplitude 3 and frequency 2 Hz is y = 3 sin(4πt), which repeats every 0.5 seconds.