Graph PlottersEnter a transformed sine or cosine equation to find its amplitude, period, phase shift, midline, and graph together.
Hover to inspect coordinates. Drag the graph to pan the view or use the wheel and zoom controls to explore the function.
ABOUT THE TOOL
For y = A sin(Bx + C) + D, the phase shift is −C/B when B is not zero. The value A controls the amplitude, B controls the period, and D moves the midline vertically. The calculator shows those relationships beside the graph.
HOW TO USE IT
A phase shift moves a periodic graph horizontally. In the standard form y = A sin(Bx + C) + D or y = A cos(Bx + C) + D, the horizontal shift is −C/B when B is not zero. This phase shift calculator lets you type the full equation or change each coefficient separately, so you can see the relationship instead of relying on a memorized rule. The live coordinate cursor makes it easier to compare the original position of a key point with its new position.
Phase shift is only one part of a transformed sine or cosine graph. The absolute value of A gives the amplitude, B controls the period, and D sets the midline. In radians, the period is 2π/|B|; in degrees, it is 360°/|B|. Change the angle unit when your worksheet uses degrees, or stay in radians for algebra and calculus. You can enter any finite value directly in the number fields.
Use this online phase shift calculator to check a trig equation, identify a horizontal movement, or prepare a graph for class. Start with sine or cosine, update the equation, and read the result cards below the graph. Zoom controls help when a high frequency creates many cycles. The tool runs in your browser, so your equation stays local unless you choose to copy a shareable link.
For y = A sin(Bx + C) + D or y = A cos(Bx + C) + D, the four constants describe different features. The absolute value of A measures distance from the midline, B controls how quickly the cycle repeats, C moves the input horizontally, and D moves the entire curve vertically. Rewriting the inside as B(x − h) makes the horizontal movement especially clear: h = −C/B. This is why the sign in a phase-shift calculation matters and why the same equation can look different when it is factored.
Consider y = 2 sin(3x − π) + 1. The amplitude is 2, the midline is y = 1, and the period is 2π/3 radians. Factoring the input gives 3(x − π/3), so the graph shifts π/3 units to the right. Enter the equation, verify those three values in the result cards, and use the cursor to locate a crest near the shifted key points. This approach also works for cosine and for degree-mode problems when the coefficients are written in degrees.
Rewrite the inside of the sine or cosine as B(x − h). In y = 4cos(2x − π) + 3, factoring gives 2(x − π/2), so the amplitude is 4, the period is π, the phase shift is π/2 right, and the midline is y = 3. This factoring step is safer than reading the sign of C in isolation because it shows the direction directly.
After entering an equation, mark the midline and the two extreme values before studying individual points. For y = 2sin(x) − 1, the midline is −1, the maximum is 1, the minimum is −3, and one cycle lasts 2π radians. The result cards and cursor let you compare these predictions with the plotted crest, trough, and crossings.
Explore related tools as you study functions: start with the online graphing calculator to plot a general equation, then use the linear graph calculator for constant rates and the slope calculator for rise over run. When a curve meets the x-axis, check the x intercept calculator; for parabolas, compare the quadratic graph calculator and vertex calculator. Use the cubic graph calculator for polynomial end behavior, the sine graph calculator and cosine graph calculator for periodic models, and the phase shift calculator for horizontal movement. The function transformations calculator compares parent and transformed curves, while the absolute value graph calculator explains V-shaped graphs. For growth and inverse relationships, try the exponential growth graph calculator, reciprocal graph calculator, and log graph calculator. For asymptote and periodic-function practice, use the tangent graph calculator, secant graph calculator, and cosecant graph calculator to compare branches, gaps, and repeating behavior.
COMMON QUESTIONS
Write the function as A sin(Bx + C) + D or A cos(Bx + C) + D, then calculate −C/B. A negative result shifts the graph left and a positive result shifts it right.
In radians, the period is 2π divided by the absolute value of B. In degrees, it is 360° divided by the absolute value of B.
Yes. Sine and cosine use the same amplitude, period, phase-shift, and vertical-shift controls.
The sign becomes clearer after factoring Bx + C as B(x − h). The value h is the horizontal shift, so rewrite the expression before deciding left or right.
The absolute value of B still determines the period, while the negative sign reverses the horizontal orientation before the other shifts are applied.
The vertical shift D is the midline: y = D. The curve oscillates A units above and below that line.