Graphing Calculator | Function, Equation, Parametric, Point

Use our free online graphing calculator to plot functions, equations (including implicitly defined functions), parametric curves (also known as parametric equations), and points in both Cartesian and polar coordinate systems, as well as oblique coordinate systems. Easily find the x-intercepts of function graphs, and compute and graph symbolic derivatives of functions and parametric expressions.

Entering Expressions into Graphing Calculator

This advanced graphing calculator is designed to be intelligent and user-friendly. When you enter an expression, it detects its type and internally adjusts the variables accordingly (to see the adjustments, just hover your mouse over the expression type label above the relevant input box):

Note: When graphing functions or parametric expressions, if you don't specify a domain (interval), this intelligent graphing calculator automatically selects a suitable domain for accurate plotting. The default domains are:

Users can modify the endpoints of the interval as needed. However, for polar or parametric graphing, the endpoints must be finite. If infinite values are entered, the graphing calculator automatically adjusts them to finite values.

About the Point Plotter

This online point grapher (aka coordinate plotter or point plotter) is designed to plot points in a plane. You'll provide the points as ordered pairs, using a simple format described below.

Entering Points into Point Plotter

To use our online Cartesian and polar point plotter, simply enter the points (a₁,b₁),(a₂,b₂),... as a₁,b₁; a₂,b₂; ... In other words, separate the coordinates of each point by a comma and the points themselves by a semicolon (note that parentheses are excluded); blank spaces are optional. The last semicolon is optional (see the note below).

You can use constant expressions such as 1/2+sin(π/3) for point coordinates.

Example: 1,2; 3/4,4; -2,sin(π/6); (This will plot three points). If you choose the degrees mode, you can simply type in sin(30) since π/6 (radians) is equal to 30°.

Connecting Points

The point grapher allows you to connect the points with line segments to form line graphs or polygons by pressing the Connect button. This toggle button will connect the points in the focused expression box. Press it again to Unconnect the points.

Note: When connecting the points, if the last point is followed by a semicolon, the point plotter will connect it to the first point—forming a (closed) polygon.

A User-Friendly Cartesian & Polar Point Grapher

Our integrated Cartesian and polar point plotter makes graphing simple in both Cartesian and polar coordinate systems. To graph points given by ordered pairs (a,b), just enter them as a,b;

These pairs can represent points in either Cartesian coordinates or polar coordinates, and our plotter will graph them based on the coordinate system you choose. Learn how to convert between Cartesian and polar coordinates

By default, points are plotted in the Cartesian coordinate system. However, simply select the Polar checkbox, and our coordinate plotter will seamlessly switch to its built-in polar coordinate system. This means the components of your ordered pairs will be treated as polar coordinates—typically represented as (r,θ)—and plotted accordingly.

Our polar point plotter offers maximum flexibility, accepting angles (θ) in radians, degrees, or grads, which you can easily select.

In addition, the coordinate plotter supports axis rotation, letting you graph points in oblique coordinate systems for even more versatility.

Point Graph in Oblique Cartesian Coordinate System

One of the most intriguing features of our point grapher is its unique ability to rotate axes. The standard Cartesian coordinate system, typically referred to as the rectangular coordinate system, employs two perpendicular axes: one horizontal and one vertical. By rotating these axes, we create a Cartesian coordinate system where the axes can intersect at any angle. This is called an oblique Cartesian coordinate system or parallelogrammic Cartesian coordinate system. This naming is due to the parallelograms formed by grid lines, which are parallel to the corresponding axes.

While many texts use Cartesian coordinate system and rectangular coordinate system interchangeably, we specifically use the full term rectangular Cartesian coordinate system to distinguish it from the oblique Cartesian coordinate system that we introduce.

Our points plotter allows for unique exploration of how point graphs appear in this generalized Cartesian coordinate system, which we refer to as the oblique Cartesian coordinate system (or simply the oblique or parallelogrammic coordinate system).

By accommodating oblique axes, our points grapher offers fresh perspectives for data visualization and analysis, particularly in fields like physics, engineering, and geometry, and game development—where non-orthogonal coordinate systems help streamline complex problems.

Point Graph in Oblique Polar Coordinate System

In the standard polar coordinate system, the polar axis is drawn horizontally. However, our polar point grapher allows you to position it at any angle and orientation by rotating the polar axis. This enables you to plot any set of points in an oblique polar coordinate system.

About the Function Grapher

Our function grapher is unique in that it allows you to visualize the same function, say f(x), in both Cartesian and polar coordinate systems. In polar coordinates, the variable x represents the angle and f(x) represents the signed distance. Since the common notation in the polar coordinate system uses θ and r, our function graphing calculator changes f(x) to r(θ), without changing the defining function, enabling you to compare the function graphs in both coordinate systems..

Comprehensive Function Visualization

To demonstrate how a function is graphed in both the Cartesian and polar coordinate systems, this function graphing calculator accomplishes this in a unique way and with remarkable ease: simply by switching coordinate systems—by selecting/deselecting the Polar checkbox. This mathematically sound approach allows users to visualize and compare the Cartesian and polar graphs of a given function.

This polar function grapher uses a unique animation algorithm to visualize the step-by-step construction of the graph of a function in the polar coordinate system like no other grapher. With its ability to rotate radial axes, it helps you understand the polar graphing process for functions in stunning animation. This capability lets you clearly visualize the construction of polar graphs of functions from beginning to end.

Moreover, this versatile and oblique function grapher enables users to rotate axes and graph functions in oblique coordinate systems, providing a powerful all-in-one visualization tool.

About the Parametric Equations Grapher

Our parametric equations grapher (aka parametric curve grapher) draws parametric curves represented by p(t) = [f(t),g(t)] in both Cartesian and polar coordinate systems.

When graphing in the Cartesian coordinate system, this is typically expressed as p(t) = [x(t),y(t)] and in the polar coordinate system as p(t) = [r(t),θ(t)]

Unique Cartesian & Polar Parametric Grapher

Our Cartesian and polar parametric curve grapher uses a unique and easy-to-follow animation algorithm to illustrate how Cartesian and polar parametric graphs are drawn. This capability lets you clearly visualize the construction of parametric curves from beginning to end.

Additionally, our parametric graphing calculator allows you to run, pause, and resume the animation and easily control the speed of the parametric curve graphing process.

By default, parametric expressions are graphed in the Cartesian coordinate system. That is, for each t, the components of ordered pairs [f(t),g(t)] will be treated as Cartesian coordinates—typically represented as [x(t),y(t)].

However, simply select the Polar checkbox, and our parametric grapher will seamlessly switch to its built-in polar coordinate system. This means that for each t, the components of ordered pairs [f(t),g(t)] will be treated as polar coordinates—typically represented as [r(t),θ(t)]—and graphed accordingly.

Besides being the first ever polar parametric curve grapher available online, it also has the unique feature of rotating radial axes when constructing polar parametric curves from scratch.

Additionally, this parametric graphing calculator allows you to rotate axes and graph parametric equations in oblique coordinate systems, where axes can be rotated to any angle and have any orientation.

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 and F(x,y) = f(x)

Whenever you have a function that's 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 graphing calculator can handle implicit functions, which are inherently defined by equations rather than explicit expressions.

Specific Applications of the Equation Graphing Calculator

As a general equation graphing calculator and implicit function grapher, it allows you to:

Graph linear equations, which implicitly define functions, in point-slope form y-y₁ = m(x-x₁) and general form ax+by = c
Graph quadratic equations representing conic sections in the standard form (x-h)² + (y-k)² = r² and the general form Ax² + Bxy + Cy² - Dx + Ey + F = 0 which may represent a circle, ellipse, parabola, hyperbola, or some degenerate graphs.
Graph level curves, expressed as F(x,y) = c.