What is a polar graph used for?

Polar graphs suit shapes defined by distance and direction from a center, such as spirals, flower-like roses, antenna radiation patterns and orbits. Equations that are messy in x and y often become simple in r and θ.

Why does my graph show a vertical line where there should be a gap?

The plotter connects sampled points, and when a function jumps from very large positive to very large negative values near an asymptote, the connecting segment looks like a vertical line. It is a drawing artifact, not part of the function. Narrowing the window or checking where the denominator is zero confirms it.

Do Bézier curves pass through their control points?

Only the first and last ones. The inner control points act like magnets that shape the curve, and the curve always stays inside the shape formed by connecting all the control points.

f(x)

Graphing & Visualization calculators

These tools draw curves: ordinary functions y = f(x), polar curves r(θ), parametric curves defined by x(t) and y(t), and Bézier curves built from control points. Seeing a curve makes roots, turning points, asymptotes and symmetry obvious in a way a table of values does not.

5 tools · Updated October 21, 2026

Showing 5 tools

B(t)Bezier Curve CalculatorDesign curvesCubic (0,0) (1,2) (3,3) (4,0) at t = 0.35(1.3318, 1.6039)

f(x)Equation Graph GeneratorFunction plotssin(x) and cos(x), −2π to 2π4 crossings, y = ±0.7071

f(x)Inequality GrapherMore graphing & visualization toolsIs (0, 0) in y > 2x − 3, y ≤ −x + 4?In the solution region

f(x)Parametric Curve GrapherFunction plotsx = t − sin(t), y = 1 − cos(t), 0 to 2πArc length 8

r(θ)Polar Graph GeneratorFunction plotsArea of r = 1 + cos(θ), θ = 0 to 2π4.712389

Which one do I need?

I want to plot y = f(x) and see where it crosses the axisEquation Graph Generator I want to compare two or more functions on the same axesEquation Graph Generator I want to draw a rose, spiral or cardioid from r(θ)Polar Graph Generator I have x and y as functions of t, such as a projectile path or a circleParametric Curve Grapher I want points on a Bézier curve from its control pointsBezier Curve Calculator

Cartesian, polar and parametric forms

A Cartesian graph plots y as a function of x, which works well for most algebra and calculus functions but cannot draw a full circle as a single function, since a vertical line would cross it twice. The Equation Graph Generator is the right starting point for anything written as y = f(x).

Polar form describes a point by its distance r from the origin and its angle θ, which makes circles, spirals, roses and cardioids short and natural to write. Parametric form gives x and y separately as functions of a third variable t, often time, so it can trace loops, self-crossing paths and motion. Use the Polar Graph Generator and Parametric Curve Grapher for those cases.

Reading what a graph tells you

Roots are where the curve crosses the x-axis, turning points are where it changes direction, and vertical asymptotes are where it shoots off toward infinity, usually where a denominator is zero. Plotting before solving gives you a sanity check: if the graph crosses the axis twice, an algebraic answer with one root is missing something.

Graphs can also mislead. A plotter samples a finite number of points, so very fast oscillation can look like noise and a vertical asymptote can be drawn as a steep line. Zoom in on interesting regions rather than relying on one wide view.

How Bézier curves work

A Bézier curve is defined by control points: it starts at the first, ends at the last, and is pulled toward the ones in between without generally passing through them. Cubic Bézier curves, with four control points, are used in vector drawing programs, font outlines and CSS easing functions.

The Bezier Curve Calculator evaluates points with de Casteljau's algorithm, which repeatedly interpolates between control points. It is numerically stable and shows geometrically how the curve is built.

Questions people ask

A function gives exactly one y for each x, but a circle has two y values for most x values. You can graph it as two functions (the top and bottom halves), or more naturally in parametric form as x = r cos t, y = r sin t, or in polar form as r = constant.