Graph, Solve, Differentiate - Graphing Calculator

The world's most advanced Cartesian and polar graphing calculator for drawing the graphs of functions, equations (including implicitly defined functions), parametric curves and point sets. Deploying the most sophisticated Cartesian and Polar coordinate systems this graphing calculator is capable of animating polar and parametric graphs and graphing in non-perpendicular coordinate systems.

A graphing calculator is a scientific calculator that is capable of graphing mathematical expressions, such as, functions in the Cartesian or polar coordinate systems.

In addition to functions, this graphing calculator, is capable of graphing equations (including implicitly defined functions), parametric equations and point sets in Cartesian or polar coordinate systems.

Syntax by Examples
  • 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

In particular, you can use this graphing calculator to graph conic sections (in the standard 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) and also graph level curves, which are in the form F(x,y) = c.

Unlike other graphing software, this graphing calculator can also graph in non-perpendicular (or non-orthogonal) Cartesian coordinate systems, where axes need not be horizontal or vertical and can intersect at any angle.

Another unique and tremendously useful feature of this polar graphing calculator is that it can animates polar graphs of functions. In addition, it is capable of animating the graph of parametric equations in both the Cartesian and polar coordinate systems. This way the Cartesian & polar graphing calculator shows how these type of graphs are constructed progressively from a starting value to an ending value on their specified domain (interval).

Furthermore, this graphing calculator can be used to solve equations in order to find x-intercepts (zeros or roots) of a given function.

Moreover, you can use the derivative graphing calculator to calculate derivatives of the 1st and 2nd order of a given function or parametric expression and graph their derivatives.

Note: to calculate the higher order derivatives and also partial derivatives of multi-variable functions you can use the partial derivative calculator.

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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.

Instruction for Graphing Calculator

It's easy to use the graphing calculator; type in an expression (function , equation, parametric or a point set) in any expression box. The graphing calculator detects the type of the expression and graphs as you type (default) in the selected coordinate system. (Don't worry about which variable (x, y , t, θ) you use, the graphing calculator automatically changes the variables according to the expression type and the selected coordinate system.)

Tips: As you type:
  • pi is replaced by π,
  • inf (infinity) is replaced by ∞, and
  • ..t is replaced by θ.

To graph piecewise defined functions or piecewise defined parametric curves type in each piece with the corresponding subinterval as a single function or single parametric expression.

The quickest way to type dom=(0, 2π) or dom=(-∞, ∞) is by deleting the domain entirely, including dom=.

Note: Since the values of trigonometric functions depend on the angle mode you select, the graphs of expressions containing trigonometric functions will, as expected, differ as you change from RAD mode (default) to other modes.

Rotate Axis, Translate and Change Scale by using your mouse

In addition to inputting data you can also use your mouse to perform some functionality unique to this interactive graphing calculator as outlined below.

  • Click on (or near) an axis and move your mouse to rotate the axis. The graph(s) are drawn in generalized (non-perpendicular) Cartesian coordinate systems. Click again to release the axis.
  • Drag the mouse to move the coordinate system together with the graphs.
  • Double-click in the canvas to move the origin to where was clicked.
  • Hold down Alt key and click on an axis to change the scale (zoom in one direction); the point which was clicked will be labelled "1" (or "-1") and becomes the new unit for that axis.

Note: If Graph As You Type is selected, the graphs are automatically updated when performing the above, and in general, any interaction with the graphing calculator, i.e., typing, mouse operations and clicking a button. Otherwise, you have to press Graph Selected Expressions to reflect the changes you have made.

You can set the following options by pressing the ⚙ (gear) button on the top right corner of the graph canvas.

Interesting curves: Graph any of the expression under Interesting Graphs in the Cartesian or polar coordinate systems by clicking on it. For best results you may need to select Graph Fineness as "+1" or higher.

This Cartesian & polar graphing calculator is an implementation of all the following each of which can be used to graph a single type of expression as the names suggest.
Insert on the bottom of multi-input panel: