Online Implicit Function & Equation Grapher

Explore our free online equation grapher, a sophisticated tool for graphing equations in the general form G(x,y) = F(x,y), including implicit functions.

About the Equation Grapher

Our equation grapher enables you to graph equations that can contain the variables x and y on both sides. That is, equations involving two variables that are in the general form G(x,y) = F(x,y) such as 2y^2+xy = x^2+2y

An equation grapher can also graph a function y = f(x). This is a special case of the general form, where G(x,y) = y F(x,y) = f(x) Whenever you have a function that is explicitly defined as y = f(x), you can simply type the right hand side to graph the function.

Implicit Function Grapher

Being a more versatile graphing tool than an ordinary function grapher, our equation grapher can handle implicit functions because they are inherently defined by an equation. Other expressions that an equation grapher can handle include:

  • Lines: Including both the general form (ax+by = c) and the point-slope form (y-y₁ = m(x-x₁)).
  • Quadratic Equations: Those whose graphs are conic sections, such as circles, ellipses, parabolas, and hyperbolas.
  • Level Curves: The set of points where a multivariable function f(x,y) remains constant.
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Functions

Lines

1 x+1 2x

Semi-circles

√(9-x^2) -√(9-x^2)

Semi-ellipses

√(9-x^2/3) √(9-x^2/3)

Parabolas

x^2 0.5x^2-4x+1 -(0.5x^2-4x+1)

Semi-hyperbolas

√(x^2-4) -√(x^2-4)

Other graphs

√(4sin(2x)) √(4cos(2x))
Equations

Lines

y = 1 x = 1 y = x+1 x = y+1 3x + y = 2 3x - y +5 = 4x+2y-2

Circles

x^2+y^2 = 9 (x-2)^2 + (y-2)^2 = 4

Ellipses

x^2/4 + y^2/9 = 1 x^2-xy+2y^2-x-2y-8=0

Parabolas

y=x^2 y = x^2-4x+4 2x^2-4xy+2y^2-x-2y-2=0

Hyperbolas

x^2/4 - y^2/9 = 1 2x^2-5xy-4y^2+9x+9y-16=0

Other graphs

x^2 = y^2 sin(xy) = cos(xy)
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Equation Grapher: Graph equations in rectangular coordinate system.
Equation grapher: Equation graphs in rectangular Cartesian coordinate system.
Oblique Equation Grapher: Graph equations in oblique coordinate system.
Oblique equation grapher: Equation graphs in oblique Cartesian coordinate system.
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To copy or save graphs right click on the image of a saved graph below and select "Copy image" or "Save image" from the pop-up menu.

Entering Equations into Equation Grapher

It's easy to use our equation grapher: simply type an equation, for example 3xy-2y = x^2+4y into any expression box. The grapher graphs as you type (default).

Note: To graph equations of the form y = f(x)—a function— simply enter f(x). When graphing functions in the Cartesian coordinate system, the grapher plots the function on the interval (domain) dom=(-∞,∞) by default if no interval is already specified. You can change the endpoints of the interval if desired. You can also use our multi-purpose polar graphing calculator or our dedicated polar function grapher to visualize function graphs in the polar coordinate system.

Function vs. Equation

Many online resources incorrectly use the terms "function" and "equation" interchangeably. Although both represent relations (sets of ordered pairs), they are not identical.

An equation G(x, y) = F(x, y) is defined by the set of ordered pairs {(x, y) | G(x, y) = F(x, y)}, which is conceptually identical to the way a function y = f(x) is defined as a set of ordered pairs: {(x, y) | y = f(x)}. While functions are often expressed as equations (e.g., y = f(x)), not all equations define a function. For example, x^2 + y^2 = 4 represents a circle centered at the origin with a radius of 2; this is not the graph of a function because it fails the vertical line test.

How Our Equation Grapher Works

Our equation grapher employs an advanced algorithm. The algorithm scans rows of pixels on the canvas to find the zeros of f(x,y)-g(x,y) for each value of y using Newton's method. It then utilizes implicit differentiation to calculate and draw tiny tangent lines at those values that satisfy the equation. This process effectively constructs the graph with a level of precision you can control by adjusting the Graph fineness setting.

Equations in Oblique Cartesian Coordinate System

Our equation grapher’s ability to graph in oblique coordinate systems relies on our graphing software’s unique capability to plot points in oblique coordinate systems.

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