Function Grapher: Plot Cartesian & Polar Graphs (Animated)

Welcome to the world's most advanced function grapher. Utilizing sophisticated Cartesian and polar coordinate systems, this interactive function plotter enables you to graph functions in both rectangular and oblique (non-orthogonal) coordinate systems. Additionally, our polar function grapher allows you to animate the polar graphing of functions, giving you the power to run, pause, or resume the animation while controlling the step-by-step plotting speed.

About the Function Grapher

Our function grapher is unique in that it plots any given function in both Cartesian and polar coordinate systems simply by switching coordinate systems.

As a polar function grapher, it uses a unique animation algorithm to visualize the step-by-step construction of the polar graph of the function like no other grapher. With its ability to rotate radial axes, it helps you understand the polar graphing process for functions in stunning animation.

Comprehensive Function Visualization

Our function plotter makes it easy to toggle between the Cartesian graph (default) and the polar graph of a given function: Simply, check or uncheck the Polar checkbox; this enables you to visualize the same function f(x) in both coordinate systems.

Since the common notation in the polar coordinate system uses θ and r for the independent and dependent variables, our grapher automatically changes f(x) to r(θ) without altering the defining function. This enables a direct comparison of the function's graphical behavior across both coordinate systems.

Moreover, our versatile oblique function grapher enables you to rotate the axes and graph functions in a skew (oblique) coordinate system, providing a powerful all-in-one visualization tool.

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Label Axes

Translate Origin

Rotate Axes

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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)
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Your browser does not support the Canvas element, or you need to enable Javascript on your browser to use this Cartesian and polar function grapher. Below are images of the grapher.

Function Grapher - Graph functions in rectangular coordinate system.
Function grapher: Function graphs in rectangular Cartesian coordinate system.
Oblique Function Grapher - Graph functions in oblique coordinate system.
Oblique function grapher: Function graphs in oblique Cartesian coordinate system.
Polar Function Grapher - Graph functions in polar coordinate system.
Polar function grapher: Function graphs in polar coordinate system.
Oblique Polar Function Grapher - Graph functions in oblique polar coordinate system.
Oblique polar Function grapher: Function graphs in oblique polar coordinate system.
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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.

Entering Function Expressions

To explore graphs of functions in the Cartesian or polar coordinate system, type a function into any expression box, for example, f(x), or r(θ). When you enter an expression, the grapher 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). The grapher will update the graph as you type (default) in the selected coordinate system.

The function plotter appends a suitable interval to function expressions and graphs them on the specified domain:

You can switch between Cartesian and polar graphs by checking or unchecking the Polar checkbox. This action will redraw the graph of the function accordingly.

How Our Polar Function Grapher Works

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

  1. 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 plotter computes the signed distance r(θ) and locates that point along the radial axis.
  2. 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.

This way, you can watch your polar graphs take shape in real time!

Insert on the bottom of multi-input panel: