Graphing Calculator: Functions, Equations, Parametric Curves, and Points X-Intercepts and Symbolic Derivative Tools

Explore our advanced, free online graphing calculator to graph 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 the specialized capability of graphing in an oblique (skew) coordinate system. Additionally, you can easily find the x-intercepts of function graphs, and calculate and graph symbolic derivatives of functions and parametric equations.

Quick Start

Enter expressions using the demonstrated syntax (blank spaces are optional):

Points: 1,2; -2,2/3; sin(π/3),cos(π/3) (Note: Enter each point (a,b) as a,b without parentheses. Separate multiple points with a semicolon.)
Functions: f(x) = x^2sin(x) + 2x + 1
Functions (custom polar notation): r(θ) = 3sin(θ)
Parametric equations: p(t) = [sin(t),cos(t)] (brackets optional)
Equations: x^3 - xy + 2y^2 = 5x + 2y + 5

No need to worry about which variable (x, y, t, θ) to use—the graphing calculator detects the type of expression as you type, and intelligently adjusts variables internally, based on the expression type and the chosen coordinate system. More on syntax

Functionality:

Select the Polar checkbox to graph in the polar coordinate system.
Press » to graph multiple expressions in the chosen coordinate system.
To visualize the progressive drawing of a polar graph or parametric curve in focus from start to end, press the Animate or ► buttons. Visualizing graph construction by animation
To find the x-intercepts of the graph of a function in focus, press the Solve button, which solves f(x) = 0—solutions are the x-intercepts. Finding x-intercepts
To calculate the symbolic derivatives (up to the second order) for a function or parametric expression in focus, press the Derivative button. Calculating symbolic derivatives and graphing

Tips - As you type:

pi is replaced by π.
..t is replaced by θ. (You can also use x or t; they are internally replaced by θ).
inf (infinity) is replaced by ∞.

More tips

About the Graphing Calculator

Our online graphing calculator, utilizing the most sophisticated Cartesian and polar coordinate systems, consolidates the following graphing tools:

Points Plotter: Easily plot points in Cartesian and polar coordinate systems. More about the point plotter.
Function Grapher: Plot Cartesian and polar graphs of functions and animate their polar curves to visualize their step-by-step construction. More about the function grapher.
Parametric Curve Grapher: Graph Cartesian & polar curves of parametric equations step-by-step with animation. More about the parametric grapher
Equation Grapher: Graph equations—including implicit functions, of the form G(x,y) = F(x,y), which can contain the variables x and y on both sides. More about the equation grapher

The graphing calculator effortlessly determines x-intercepts (also known as zeros or roots) of a function. As a derivative graphing calculator, it computes and graphs symbolic derivatives up to the second order for both functions and parametric equations.

Animation and Visualization

To demonstrate how graphs of functions are constructed in the polar coordinate system, as well as graphs of parametric equations in both Cartesian and polar coordinate systems, our graphing calculator utilizes a sophisticated and uniquely interactive animation method.

This feature automatically draws these types of graphs step-by-step, giving you full control over the animation, allowing them to run, pause, resume, and adjust the animation speed through a convenient and intuitive interface.

Our graphing calculator also enables you to rotate axes and in an oblique coordinate system, where the axes can be rotated and intersect at any angle—not just 90 degrees.

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):

Function: If you use x as the independent variable, the function graphing calculator will label the expression as f(x)= or f(θ)= depending on the coordinate system.
Equation: If your expression includes an equal sign (=), the calculator switches to equation mode, and the equation graphing calculator (also known as implicit function grapher) will display Eq: in the label.
Points: If your expression uses both commas and semicolons, the calculator switches to points mode, and the points plotter will display x,y; or r,θ;, depending on the coordinate system, for the expression label.
Parametric curve: A single comma indicates a parametric expression, switching the calculator to parametric mode. The parametric curve graphing calculator (also known as parametric equations grapher) displays p(t)= for the expression label, and replaces x with t in the expression. Deleting the comma switches back to function mode, replacing t with the appropriate variable.

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:

dom=(-∞,∞) for graphing functions in the Cartesian coordinate system.
dom=(0,2π) for graphing functions in the polar coordinate system.
dom=(0,2π) for graphing parametric curves in both Cartesian and polar coordinate systems.

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 free online Cartesian and polar 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. This unique point plotter allows users to rotate axes, enabling the graphing of points in oblique coordinate systems.

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.

About the Function Grapher

Our unique, interactive online function grapher enables users to explore graphs of functions in the Cartesian coordinate system—both rectangular and oblique. As a polar function grapher, it also allows you to visualize functions in the polar coordinate system, and animate the process of polar graphing of functions with stunning clarity.

Our function graphing calculator 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 grapher changes f(x) to r(θ) and vice versa, depending on whether the Polar checkbox is selected. This is done without changing the defining function, enabling you to compare the function graphs in both coordinate systems.

Entering Function Expressions into the Function Grapher

For Cartesian graphs of functions use x as the independent variable. For polar graphs of functions use θ. To type θ in a function expression, type "..t". You can also simply use t or even x. They will internally be replaced by θ.

The function graphing calculator graphs on a specified interval (domain). If no interval is specified, the grapher appends a suitable interval to function expressions. It uses dom=(-∞, ∞) for graphing functions in the Cartesian coordinate system; you can change these endpoints if needed.

On the other hand, for polar graphing, it uses dom=(0, 2π), and while you can change the endpoints, they must be finite for polar graphing of functions. The polar function grapher automatically adjusts infinite values to some finite ones.

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 via 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.

How Our Polar Function Grapher Works

This unique interactive polar function graphing calculator plots functions r(θ) directly in the polar coordinate system, similarly to how you would graph them on paper—without converting to Cartesian coordinates.

For each value of angular coordinate θ, a temporary radial axis is drawn, making an angle of θ with the polar axis. The polar function graphing calculator computes the signed distance r(θ) and locates that point along the radial axis.
It then connects this point to the next point located using the same method with a slightly larger value of θ. This continues until the complete polar graph of the function is drawn.

Our polar grapher also offers an animated graphing process, as detailed below.

Polar Function Graph Animator

Why animation? Polar curves can be intricate, often featuring multiple loops. Most other graphers display the polar graph of a function instantly, without showing where it starts or ends, or how any loops—if present—are traced.

To address this, it's crucial to draw a polar graph step-by-step, allowing for a clear visualization of its creation on its domain. Our polar graph animator, equipped with a sophisticated polar coordinate system, is specifically designed for this.

It is the first to introduce the proper method for graphing functions in the polar coordinate systems through a controlled animation.

About the Parametric Equations Grapher

Our online parametric equations grapher (aka parametric curve grapher) is not only capable of drawing graphs of parametric equations in Cartesian coordinate systems, but it's also the only polar parametric grapher available, which allows you to visualize parametric equations in the polar coordinate system. This unique, interactive graphing tool also allows you to animate the process of parametric graphing in both coordinate systems, including oblique ones.

Our parametric equations grapher draws parametric curves represented by p(t) = [f(t),g(t)]

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)]

Entering Parametric Expressions into the Parametric Graphing Calculator

To explore the graph of parametric equations, type a parametric expression—typically expressed as p(t) = [f(t),g(t)]—in any expression box. For example, you could type [sin(t),cos(t)]. Using the enclosing brackets [ ] is optional. The parametric graphing calculator graphs as you type (by default) in the user-selected Cartesian or polar coordinate systems.

Our parametric equations graphing calculator graphs on a specified interval (domain). If no interval is specified, the grapher appends dom=(0, 2π). You can change the endpoints, but they must be finite. The parametric grapher automatically replaces infinite values with finite ones.

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.

In addition to 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.

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 sand 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.

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

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.

Specific Applications of the Equation Graphing Calculator

As a general equation graphing tool 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.