Equation Grapher | Implicit Function 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 an 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.

Being a more versatile graphing tool than a 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 the equation of a line in both general form (ax+by = c) and point-slope form (y-y₁ = m(x-x₁)), quadratic equation—whose graphs are conic sections (circles, ellipses, parabolas, and hyperbolas)— and level curves, for example.

Try our Graphing Calculator, which in addition to graphing equations and functions can graph parametric equations and points.

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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.

Instructions for Using the Equations Grapher

Tips - as you type:

  • pi is replaced by π.
  • inf (infinity) is replaced by .

MouseMatics: Find out how to use your mouse to rotate axes, change scales, and translate coordinate systems.

Entering Equations into Equation Grapher

It's easy to use our equation grapher; type in an equation, for example 3xy-2y = x^2+4y in 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 polar graphing calculator to visualize function graphs in the polar coordinate system.

The quickest way to type dom=(-∞,∞) is to delete the domain entirely, including dom=.

Function vs. Equation

Many online resources incorrectly use "function" and "equation" interchangeably. However, these terms are not identical.

Although functions are expressed as equations, i.e., y = f(x), not all equations define a function. As an example, x^2+y^2 = 4, whose graph is a circle centered at the origin with a radius of 2, is not the graph of a function since it does not pass the vertical line test.

How Our Equation Grapher Works

Our equation grapher employs an advanced algorithm. This algorithm starts by investigating rows of pixels on the canvas to find the zeros of f(x,y)-g(x,y) for each value of y, employing Newton's method. It then uses implicit differentiation to draw tiny tangent lines at those values that satisfy the equation. This process effectively builds the graph.

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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