Innovative Polar Function Grapher
Visualize functions with unmatched visual clarity and interactivity using our unique polar function grapher. This advanced online polar function grapher utilizes a specially designed 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 through engaging animation. Pause, resume and change the speed of animation with the provided slider.
Comprehensive Function Visualization
To demonstrate how the same function is graphed in the Cartesian coordinate system, this grapher can also draw its Cartesian graph. It accomplishes this in a unique way and with remarkable ease, simply by switching coordinate systems. This mathematically most proper approach allows you to visualize and compare the polar graph and Cartesian graph of a given function in either coordinate system. Furthermore, this grapher empowers you to explore the unique world of graphs of functions in oblique coordinate systems, where you can independently rotate axes—all in one powerful tool.
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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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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.
Tips - 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 ∞.
Note: Our graphing software allows you to use constant expressions, such as π or 1+√(2) wherever you can use a literal number.
Instructions for Function Grapher (Cartesian and Polar)
This interactive online function grapher can be used to graph 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 graphs on a specified interval (domain). If no interval is specified, the grapher appends a suitable interval to function expressions. It uses dom=(-∞, ∞) for functions in the Cartesian coordinate system, and dom=(0, 2π) in the polar coordinate system. 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.
You can switch between polar and Cartesian coordinate systems by checking or unchecking the Polar checkbox. This will redraw the graph of the function as the polar graph or Cartesian graph accordingly.
The polar grapher 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.
This polar function grapher allows you to watch the entire animated graphing process, as detailed below.
Using Polar Graphing Animation Feature
This feature allows you to visualize the step-by-step process of creating a polar graph. To activate the animation feature, press ► at the bottom of the 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 θ1 and ending at θ2. It shows whether any loops or parts of the polar curve are traced multiple times.
Controlling Animation
- To pause the animation, press ‖.
- To stop the animation, press Done. This also closes the animation interface. To display the animation interface again, press the Animate button.
- Use the slider to adjust the animation speed.
Additional Options
- 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.
Remark: To graph piecewise defined functions type in each piece with the corresponding subinterval as a single function.
Graph Accuracy Setting
The quality of the resulting graph is controlled by the Graph Fineness setting. A higher graph fineness results in a more accurate graph, but it also takes longer to graph the equation.
Labeling Axes
To label an axis, click on the ▼ icon at the top right of the canvas. Type in any number for which you want to label the axes, and press the Label button. You can also use constants like π, or even constant expressions such as 1+3π/2.
Rotating Axes
Our graphing software offers a unique feature: axis rotation. This allows you to visualize the graph of a given function in the oblique (non-perpendicular) coordinate system. To rotate an axis, click on the ▼ icon at the top right of the canvas, and then enter the angles by which you want to rotate an axis and press the Rotate button. The axis will rotate and the graphs will be redrawn to reflect the rotation.
Copying & Saving Graphs
- Click the Copy/Save graph button. This will create a copy of all the graphs on the canvas (graphing area) which will appear as an image below the graphing calculator..
- Right-click on the image and select the appropriate option from the context menu. Depending on your device and preferences, you might be able to:
- Copy: Create a copy of the image to your clipboard for pasting elsewhere.
- Save image as... : Save the image as a file on your device in a chosen format (e.g., PNG, JPG).
Evaluating Functions
- To evaluate a function, type a number or constant expression in the provided box.
- 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. For best results, you may need to select Graph Fineness as "+1" or higher.
Mouse Operations in Our Graphing Software
Our graphing software allows you to use your mouse to perform unique operations as outlined below.
How to Rotate Axes
To rotate an axis, hold down the Alt key and click on or near the axis. This will select the axis, and its color will change to red. Move the mouse. The selected axis will rotate accordingly, and the graphs are redrawn to reflect the rotation of the axis. Click again with the Alt key pressed, to release the axis.
To restore the coordinate system to its normal state press the Reset button (if hidden, press the ▼ icon).
How to Change Scales
To change the scale (zoom in one direction), hold down the ctrl key and click on an axis. The point that was clicked will be labelled "1" if clicked on the positive side, or "-1" if clicked on the negative side of the axis; it becomes the new unit for that axis.
How to Translate the Coordinate system
You can move the coordinate system in one of the following ways:
- Click on the canvas and drag the mouse; this will translate the origin and the graphs together.
- Double-click on the canvas to move the origin to the location where you clicked; the graphs are updated.
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.
- Select the angle mode (radians - default, degrees, grades).
- If you deselect the Graph as you type option, you will have to press Graph selected expressions, which then appears at the bottom of the calculator, to update the graphs whenever you make any changes to the expressions or the coordinate plane (i.e., move the origin, rotate axes, etc.).
- You have the option to display controls that will automatically rotate the axes.
- The function and equation graphing calculator remembers the expressions so that you can use them in future visits. You can press Reset Calculator to clear them.
What is a Polar Function Grapher?
A polar function grapher is a specialized calculator that draws the graph of a function on a given domain in the polar coordinate system. This graph is called the polar graph or the polar curve of the given function.
The process of graphing in the polar coordinate system and rendering it with a function graphing calculator is fundamentally different from graphing in the Cartesian coordinate system. This is because it's essential to draw a polar graph step-by-step to allow visualization of how it is created on its domain.
Our polar function grapher, equipped with comprehensive Cartesian & polar coordinate systems, is designed for this purpose. It helps visualize how a polar curve is progressively constructed by means of polar graphing animation, with which you can control the speed of the step-by-step animation with exceptional clarity. This way, you can watch how all your cool polar graphs are drawn.
This interactive polar function grapher has been developed to graph, and particularly, to show by animation how the graph of a function r = f(θ) is created in the polar coordinate system. Polar curves can be very complicated and may have many loops. Other polar graphers display the polar graph of a function out of the blue, not showing where the curve starts or ends, and whether or how the loops, if any, are traced. This unique polar function graphing calculator introduces the proper way for graphing functions in the polar coordinate systems. Namely, it starts graphing from the initial value of an angular coordinate, θ₁, and progressively shows the graphing process up to the final value of θ₂, showing whether the loops or any part of the curve re-traced. Moreover, this polar function grapher enables you to change the speed of the polar graphing process.
How Our Polar Function Grapher Works
This unique polar function grapher plots graphs of a function r(θ) in the polar coordinate system, similarly to how you would graph them on paper.
- 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.
