# Function Grapher | Cartesian & Polar (Animated Graph)

Welcome to the world's most advanced function grapher. Utilizing the most sophisticated Cartesian and polar coordinate systems, this unique interactive online function grapher enables you graph functions in the Cartesian coordinate system (both rectangular and obliquenon-orthogonal). Additionally, it allows you to visualize functions in the polar coordinate system, and animate the process of polar graphing of functions with stunning clarity.

Our function grapher is unique in that it plots your function, say `f(x)`, in the commonly used Cartesian coordinate system. Additionally, it allows you to visualize the same function in the polar coordinate system, where 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 grapher changes `f(x)` to `r(θ)`, without changing the defining function, enabling you to compare the function graph in both coordinate systems..

Our function grapher makes it easy to switch between Cartesian graph and polar graph of a given function. Simply, check or uncheck the Polar checkbox, to draw the function in the corresponding coordinate system.

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.

## How Our Polar Graphs of Function Constructed

Our polar function grapher plots graphs of functions in the polar coordinate system, similarly to how you would graph them on paper, showing step-by-step construction using a sophisticated animation algorithm developed by A.M.I.R.

For each value of `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.

The polar function grapher then connects this point to the next point located using the same method with a slightly larger value of `θ`. The online polar function graphing calculator thus completes the polar graph of the given function.

Our polar grapher allows you to watch the entire animated graphing process, as detailed in Using Polar Graphing Animation Feature.

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Label Axes

Rotate Axes

ResultsHide
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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Graph Thickness
Angle Mode
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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 context (pop-up) menu.

As you type:
• ..t is replaced by `θ`. (You can also use `x` or `t`; they are internally replaced by `θ`).
• pi is replaced by `π`.
• inf (infinity) is replaced by `∞`.
More tips

MouseMatics: Learn how you can use your mouse to rotate axes (graph in oblique coordinate system), change scales, and move coordinate system.

## Instructions

Our interactive online function grapher can be used to graph functions (including piecewise defined functions ) in the Cartesian (both rectangular and oblique), and polar coordinate systems. In particular, this unmatched function grapher is designed to show, in a clear and intuitive way, how polar graphs are created, making it ideal for teaching or learning about polar graphing.

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

The function grapher appends a suitable interval to function expressions and graphs them on the specified domain. For Cartesian graphs it appends dom=(`-∞`, `∞`), and for polar graphs it appends dom=(`0`, `2π`). You can change the endpoints, but they must be finite for graphing functions in the polar coordinate system. The polar function grapher automatically changes infinite values to finite ones. It uses a sophisticated interactive animation method to draw the polar graph on the specified domain, dom=(`θ1`, `θ2`), allowing you to see the rotating radial axes and radial distances.

### Using Polar Graphing Animation Feature

To activate the animation feature, press at the bottom of the polar function grapher (if hidden, press Animate first).

• The polar grapher starts the polar graphing animation for the focused function. The animation progressively draws the graph, starting at angle `θ1` (from the polar axis and ending at `θ2`. It shows whether any loops or parts of the polar curve are traced multiple times. You can see the rotating radial axes by checking the corresponding checkbox (selected by default).
• You can press to pause the animation or Done to stop it. This also closes the animation interface. To display the animation interface again, press Animate.
• Use the slider to adjust the animation speed.
• You can optionally show or hide the rotating radial axes by checking or unchecking the Show radial axes checkbox (by default, it's checked).

### Graphing Multiple Functions

To graph multiple functions, press the » button to show the multi-graph pane with expression panels.

• Add or delete panels with the + or × buttons if necessary.
• Select or deselect checkboxes to show or hide graphs.

### Graph Accuracy Setting

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

### Copying or Saving Graphs

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

### Evaluating Functions

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

### Interesting Curves

Interesting curves: Graph any of the expressions under Interesting Graphs 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):

• Change graph thickness using the slider.