What is the difference between a function and an equation?

While functions are often expressed as equations, not all equations define a function. For example, a circle is defined by an equation but fails the vertical line test, meaning it is not a function.

Equation Grapher | Implicit Function Plotter | Graph Equations with Variables on Both Sides

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 plotter can also graph a function y = f(x). This is a special case of the general form, where G(x,y) = y and 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. You can use our implicit function grapher to graph, for example:

Lines: Including both the general form (ax+by = c) and the point-slope form (y-y₁ = m(x-x₁)).

Special Equation Types

Other expressions that our equation grapher can handle include:

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.

Animate Axis Rotation: x y ► ⬛

message

⌫

f( ) =

Type an equation

⌨

4 Decimal places

Label Axes x-Axis: y-Axis:

Translate Origin O_x: O_y:

Rotate Axes x-Axis°: y-Axis°:

⚙

caption

...

Slow Fast

Show angular axes Done

Disable Soft Keyboard

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 Our Equation Grapher

MouseMatics: Did you know? You can use your mouse to rotate axes, change scales, and translate the origin.

Tips - While typing:

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

More tips

Entering Equations into the Equation Grapher

Using our equation grapher is simple: just type an equation—for example, 3xy-2y = x^2+4y—into any expression box. By default, the tool graphs your equation as you type.

Note: To graph equations of the form y = f(x)—a function— use our multi-purpose graphing calculator, which graphs functions faster. There, you can simply enter f(x) and specify a domain (interval). It also supports graphing 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 and implicit function 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.