Sine Function Grapher

About the Sine Function Grapher

Plot y = A·sin(Bx + C) + D and explore amplitude, period and phase shift.

How to use this calculator

1. Enter the values of A, B, C, and D in the function y = A·sin(Bx + C) + D.
2. Check the graph to see how the wave moves and stretches.
3. Read the displayed amplitude, period, and phase shift to understand the shape of the curve.
4. Use the example and the formula to compare your function with a standard sine graph.

The formula explained

The tool graphs functions of the form y = A·sin(Bx + C) + D, which describes a sine curve stretched by A, compressed by B, shifted horizontally by C, and shifted vertically by D. It also helps identify the amplitude |A| and period 2π/|B|.

• A = amplitude factor, controls vertical stretch or reflection
• B = affects the period, since period = 2π/|B|
• C = horizontal shift inside the sine function
• D = vertical shift, moving the graph up or down
• x = input value on the horizontal axis
• y = output value on the vertical axis

Step by step method

1. Start with the parent function y = sin(x), which has amplitude 1 and period 2π.
2. Apply the vertical changes first: multiply by A to stretch or reflect, then add D to move the graph up or down.
3. Use B and C to find the period and phase shift, because the inside of the sine changes how quickly the graph repeats and where it starts.
4. Sketch one cycle using key points such as the midline, maximum, and minimum to check your graph.

Worked example

Problem. Graph y = 2·sin(3x - π) + 1 and identify its amplitude, period, and phase shift.

1. Amplitude is |2| = 2, so the graph goes 2 units above and below its midline.
2. Period is 2π/|3| = 2π/3, so one full cycle repeats every 2π/3 units.
3. Phase shift comes from 3x - π = 3(x - π/3), so the graph shifts right by π/3, and the midline is y = 1.

Tips and common mistakes

• If A is negative, the sine graph is reflected over its midline, not over the x-axis.
• Be careful with phase shift, rewrite Bx + C by factoring out B before identifying the horizontal shift.

Frequently asked questions