Graphing Calculator | Function, Equation, Parametric, Points

Welcome to our world's most advanced free online graphing calculator, ideal for graphing the most common types of mathematical expressions — functions, equations, parametric curves, and point sets. It can also find the x-intercepts, calculate and graph symbolic derivatives, and graph in oblique coordinate systems.

Our online graphing calculator is a sophisticated and feature-rich graphing software application for drawing the graphs of functions, equations (including implicitly defined functions), parametric curves and points in the Cartesian and polar coordinate systems.

Here are some examples of syntax:
  • f(x) = x^2sin(x) + 2x + 1 (function)
  • x^3-xy+2y^2 = 5x+2y+5 (equation)
  • p(t) = [sin(t), cos(t)] (parametric)
  • 1,2; -2, 2/3; sin(π/3), 2^3-1 (points)
More on Syntax

It can easily find the roots of a function (also known as zeros or x-intercepts), and also calculate symbolic derivatives of functions and parametric curves and graph the derivatives.

This graphing calculator has features that enable you to animate the graphing process in a unique way that helps you understand it better. It is also unique in its ability to visualize graphs in an oblique coordinate system where each axis can be rotated independently. These features provide an interactive way to learn about graphing.

In particular, you can use this graphing calculator to:

  • Graph linear functions and linear equations in point-slope form and slope-intercept form.
  • Graph conic sections in the standard form such as (x-h)^2 + (y-k)^2 = r^2, and the general form (Ax^2 + Bxy + Cy^2 - Dx + Ey + F = 0), which can be a circle, ellipse, parabola, hyperbola, or some degenerated graphs.
  • Graph level curves, which are in the form F(x,y) = c.
  • Solve equations to find x-intercepts (zeros or roots) of a given function.
  • Calculate and graph the symbolic derivatives of the 1st and 2nd order of a given function or parametric expression.
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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))
Functions – Polar

Lines

2csc(θ) 2sec(θ) 1/(sin(θ) - cos(θ))

Circles

1 2 6sin(θ) 8cos(θ)

Spirals

θ θ/5 dom=(0, 10π) √(θ) dom=(0, 10π) 1/θ dom=(0, 10π)

Roses

4sin(3θ) 4sin(2θ) 4sin(5θ) 4sin(4θ)

Ellipses

1/(1-.8cos(θ)) 1/(1-.8sin(θ)) 1/(1+.8cos(θ)) 1/(1+.8sin(θ))

Parabolas

1/(1-sin(θ)) 1/(1+cos(θ)) 1/(1+sin(θ)) 1/(1-cos(θ))

Hyperbolas

1/(1+2cos(θ)) 4/(1+2sin(θ)) 1/(1-2cos(θ)) 4/(1-2sin(θ))

Cardioids

3+3cos(θ) 2+2sin(θ) 3-3cos(θ) 2-2sin(θ)

Limacons

2+3cos(θ) 1+2sin(θ) 2-3cos(θ) 1-2sin(θ)

Lemniscates

√(4sin(2θ)) √(4cos(2θ))

Butterfly curve

e^sin(θ)-2cos(4θ)+sin((2θ-π)/24)^5 dom=(0, 12π)
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 24x^2-50xy-49y^2+97x+93y-164=0

Other graphs

x^2 = y^2 sin(xy) = cos(xy)
Equations — Polar
Currently, not available.
Parametric

Lines

[t, 1] dom=(-5, 5) [1,t] dom=(-5, 5) [t, 2t] dom=(-5, 5)

Circles

[4sin(t), 4cos(t)] [3sin(t)+1, 3cos(t)+1]

Ellipses

[4cos(t), 3sin(t)] [3cos(t), 4sin(t)] [4sin(t), 3cos(t)] [3sin(t), 4cos(t)]

Parabolas

[t, t^2] dom=(-4, 4) [t^2, t] dom=(-4, 4)

Hyperbolas

[3sec(t), 4tan(t)] [3tan(t), 4sec(t)]

Other parametric graphs

[5sin(t), 4cos(t)] [5sin(t), 4cos(2t)] [5sin(t), 4cos(3t)] [5sin(2t), 4cos(t)] [5sin(2t), 4cos(3t)] [5sin(2t), 4cos(5t)] [5sin(3t), 4cos(5t)] [5sin(3t), 4cos(7t)] [5sin(5t), 4cos(7t)] [5sin(7t), 4cos(9t)]

Butterfly curve

[sin(t)(e^cos(t)-2cos(4t)-sin(t/12)^5), cos(t)(e^cos(t)-2cos(4t)-sin( t/12 )^5)] dom=(0, 12π)
Parametric – Polar

Lines

[2csc(t), t] [2sec(t), t] [1/(sin(t) - cos(t)), t]

Circles

[1, t] [2, t] [6sin(t), t] [8cos(t), t]

Spirals

[t, t] [t/5, t] dom=(0, 10π) [√(t), t] dom=(0, 10π) [1/t, t] dom=(0, 10π)

Roses

[4sin(3t), t] [4sin(2t), t] [4sin(5t), t] [4sin(4t), t]

Ellipses

[1/(1-.8cos(t)), t] [1/(1-.8sin(t)), t] [1/(1+.8cos(t)), t] [1/(1+.8sin(t)), t]

Parabolas

[1/(1-sin(t)), t] [1/(1+cos(t)), t] [1/(1+sin(t)), t] [1/(1-cos(t)), t]

Hyperbolas

[1/(1+2cos(t)), t] [4/(1+2sin(t)), t] [1/(1-2cos(t)), t] [4/(1-2sin(t)), t]

Cardioids

[3+3cos(t), t] [2+2sin(t), t] [3-3cos(t), t] [2-2sin(t), t]

Limacons

[2+3cos(t), t] [1+2sin(t), t] [2-3cos(t), t] [1-2sin(t), t]

Lemniscates

[√(4sin(2t)), t] [√(4cos(2t)), t]

Other parametric graphs

[5sin(t), 4cos(t)] [5sin(t), 4cos(2t)] [5sin(t), 4cos(3t)] [5sin(2t), 4cos(t)] [5sin(2t), 4cos(3t)] [5sin(2t), 4cos(5t)] [5sin(3t), 4cos(5t)] [5sin(3t), 4cos(7t)] [5sin(5t), 4cos(7t)] [5sin(7t), 4cos(9t)]
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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.

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

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

Instructions for Our Graphing Calculator

Our graphing calculator is a powerful tool for drawing the graph of functions, equations, parametric curves and point sets. Simply type the expression into any expression box, and the software will automatically detect the type of expression and generate the graph in real-time (default) using Cartesian or Polar coordinate systems. You can easily switch between them by selecting/deselecting the Polar checkbox. The software is designed to automatically adjust the variables according to the expression type and coordinate system, so you don't have to worry about which variable (x, y, t, or θ) to use.

Our graphing tool is designed to be intelligent and user-friendly. When you enter a function using x for its independent variable, the function graphing calculator automatically changes the expression label to f(x) = or f(θ) =, depending on the selected coordinate system. If your expression contains the equal sign (=), the calculator will switch to equation mode. The equation graphing calculator then displays "Eq:" in the expression label. If your expression contains both commas and semicolons, the graphing calculator will switch to points mode and the points graphing calculator changes the expression label to x, y; or r, θ;, depending on the coordinate system. If your expression contains only a comma, the graphing tool will switch to parametric mode and the parametric graphing calculator displays p(t) = in the expression label, replacing all x's in the expression with t's. If you delete the comma, the software will switch back to function mode and replace all t's with x's or θ's, depending on the coordinate system. To see these changes, simply click anywhere on the calculator.

For convenience, the graphing calculator appends a suitable interval to the function and parametric expressions. The software then graphs the expressions on that specified domain. It appends dom=(-∞, ) to function expressions when graphing in the Cartesian coordinate system, and otherwise dom=(0, ). You can change the end-points of the interval if desired. However, the end points must be finite for polar or parametric graphing. The calculator will automatically change any infinities to finite values.

Using Polar and Parametric Graphing Animation Feature

Our polar and parametric graph animator uses a sophisticated interactive animation method to draw function graphs in the polar coordinate system as well as parametric graphs in both Cartesian and polar coordinate systems. It shows whether there are any loops, or whether any parts of the polar or parametric curve are traced multiple times.

To activate the animation feature, press the button at the bottom of the graphing tool (if hidden, press Animate first).

Graphing Multiple Expressions

To graph multiple functions, equations, parametric curves or point sets, press the » button to show the multi-graph pane with expression panels, and type in your expressions in any expression field.

Graph Accuracy Setting

Select Graph Fineness option for the desired curve accuracy. Higher accuracy takes longer to graph.

Copying & Saving Graphs

  1. Click the Copy/Save graph button.
  2. A copy of the graphs will appear below the graphing calculator.
  3. Right-click on the image and select the appropriate option from the context menu.

Evaluating Functions & Parametric Expressions

To evaluate a function or parametric expression, type a literal number or numerical expression in the provided box. The calculator will display the calculated value, rounded to the number of decimal places set by the slider.

Finding X-Intercepts

Our calculator can be used as an equation solver to find the x-intercepts (also known as zeros or roots) of a function. To do this, press the Solve button. The equation solving calculator will then find the x-intercepts of the function in focus by solving the equation f(x) = 0 and display the x-intercepts on the screen. Notes on finding x-intercepts

Calculating and Graphing Symbolic Derivatives

In addition, the calculator can be used as a derivative graphing calculator. To calculate the first and second order derivatives of a function or parametric expression in focus, press the Derivative button.

After the derivatives are displayed, you can press Graph f, f' or Graph f, f', f'', which also appear on the screen, to draw the graphs of the function or parametric expression and their derivatives in a new window. You can also add the calculated derivatives to new panels by selecting them (selected by default). These panels will be appended to the bottom of the multi input pane. This will allow you to use them to find, for example, the critical points of the function by pressing the Solve button as described above. Press the OK button to close the derivative window. Find out more about first & second order derivatives.

Interesting Curves

Graph any of the predefined expressions under the Interesting Graphs selections, located on the multi-input pane, to render some cool Cartesian and polar graphs by selecting it. For best results, you may need to select Graph Fineness as "+1" or higher.

Settings

Press the ⚙ (gear) button to set options (if the button is hidden, first click on the icon at the top right of the canvas):

Insert on the bottom of multi-input panel: