# Graphing Calculator - Function, Equation, Parametric, Point

The most advanced and easy-to-use 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 the 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 graphing calculator is that it can animate 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 graphing calculator shows how these types 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 this 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.)

• To graph two or more expressions on the same Cartesian or polar coordinate system press » to display the multi-graph pane. The multi-graph pane consists of expression panels, which can be added or deleted as desired by pressing + or × on each panel, respectively. Selecting or deselecting the checkbox for any expression displays or hides the corresponding graph.
• For convenience, the graphing calculator appends a suitable interval, dom = (0, 2π) or dom = (-∞, ∞), respectively, to the function and parametric expressions and graphs on that specified domain. You can change the end points if desired. The end points must be finite for polar or parametric graphing. The graphing calculator automatically changes infinities, if any, to finite values. The graphs are drawn as you type.
• You can Animate the polar graph of a function or parametric curves (in both the Cartesian and polar coordinate systems) to see how they are constructed.
• You can use this tremendously useful feature by pressing at the bottom of the graphing calculator (if it's hidden, press the Animate button first).
• It starts the animation of the polar graphing process of the function or parametric graphing process of the parametric expression in focus. The graph is drawn progressively from the initial value to the final value on the specified domain.
• You can then press || to pause the animation or press Done to stop it. This also closes the animation interface. To display it again press the Animate button at the top of the polar grapher.
• You can also change the speed of graphing animation by using the slider provided
• You can set the fineness of the graphs by selecting the desired option from the Graph Fineness drop-down list. In general, the higher the accuracy, the longer it takes the graphing calculator to draw the graphs.
• When graphing using the Cartesian coordinate system, this Graphing Calculator is capable of graphing in non-perpendicular Cartesian coordinate systems by rotating the axes. Enter the amount of angles by which you want to rotate the axes and press Apply.
• You can also label an axis with any number (π, π/2, 1+√2, etc.).
• To copy or save graphs first press the Copy/Save graph button. An image of the graphs will appear below the graphing calculator. You can then use your browser's capability to save it or copy it in your documents.
• To evaluate a function or parametric expression type in a number or a numeric (constant) expression in the box provided; the graphing calculator displays the calculated values with the number of decimal places which can be specified by using the slider provided.
• The graphing calculator can be used as an equation solver to find the x-intercepts (zeros or roots) of a function. Press the Solve button to solve the equation f(x) = 0.
• In addition you can use the graphing calculator as a derivative calculator. Press the Derivative button to calculate the 1st and 2nd order derivatives of functions and parametric equations. You can then graph the derivatives.
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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.

• Change the thickness of the graphs by using the slider provided.