Graph PlottersPlot a transformed cosine curve and see its amplitude, period, phase shift, midline, and extrema update as you work.
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
Cosine uses the same transformation pattern as sine. A controls amplitude, B controls period, C controls horizontal position, and D moves the midline. A cosine graph begins at a maximum or minimum when C and D are zero.
HOW TO USE IT
A cosine graph is periodic like a sine graph, but its starting position makes it especially useful for studying maximums, minimums, and horizontal shifts. In y = A cos(Bx + C) + D, A controls amplitude, B controls period, C controls phase shift, and D sets the midline. This cosine graph calculator displays the equation and the key values together, so each change has an immediate visual result.
Cosine and sine use the same transformation rules. The identity cos(x) = sin(x + π/2) explains why one curve can be represented as a horizontal shift of the other. Switch between radians and degrees to match your problem, and use the coordinate cursor to examine a point at a maximum, minimum, or zero crossing. Direct number fields accept any finite amplitude, frequency, phase term, or vertical shift.
Use this online cosine graph calculator for classwork, graph transformations, and quick verification. The toolbar includes zoom, fullscreen, settings, PNG download, and share controls. The explanatory sections below the tool cover the important vocabulary in natural language, making the page useful both as a calculator and as a short reference for cosine transformations.
The parent cosine function begins at a maximum, moves through the midline, reaches a minimum, and repeats. For y = A cos(Bx + C) + D, the midline is y = D, the amplitude is |A|, and the period is 2π/|B| in radians. A negative A flips the starting crest into a trough. The phase term shifts every maximum, minimum, and midline crossing together, so marking one complete cycle is often the clearest way to check a transformed cosine equation.
Take y = −2 cos(x + π/3) + 4. The curve has midline y = 4, amplitude 2, and period 2π. Since x + π/3 = x − (−π/3), the phase shift is π/3 units left. Enter the equation, identify the starting trough at the shifted position, and use the cursor to compare the plotted extrema with the expected values 2 and 6.
For y = 3cos(2x + π) − 2, the amplitude is 3, the period is π, the phase shift is π/2 left, and the midline is y = −2. Because the coefficient of cosine is positive, the curve starts at a crest relative to its shifted position. The expected maximum is 1 and the minimum is −5, which gives two useful values to check in the graph.
The parent cosine function begins at 1 when x = 0, while sine begins at the midline. That makes cosine a natural model when a quantity starts at a high or low extreme, such as a simple seasonal cycle beginning at its peak. A phase shift can move that starting extreme, so always inspect the inside of the equation before assuming the first visible point is at x = 0.
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
Cosine is a horizontal shift of sine: cos(x) = sin(x + π/2).
In radians, divide 2π by |B|. In degrees, divide 360° by |B|.
A negative amplitude reflects the cosine graph across its midline.
For y = A cos(Bx + C) + D, the maximum is D + |A| and the minimum is D − |A|.
The parent cosine function is even, which means cos(−x) = cos(x). A horizontal phase shift can remove that symmetry around the y-axis.
Yes. A vertical shift or small amplitude can keep the entire range above or below y = 0.