The 'Graphs' pane is where you can tell the applet what to plot.

Clicking on the 'New Graph' button pops up a text box in which you can enter an equation to plot.
The simplest case would be of the form "y=..." where "..." is an expression involving "x". When you complete the entry (by pressing the Enter key) the graph will be displayed. Try this with something like "y=x^2-2x" for example. [Note: If you enter just "x^2-2x" it will assume you mean "y=x^2-2x".]

Other possibilities that are recognized include more general equations in x and y as well as equations in polar coordinates. (At present 'r' is used for the radius and 'theta' for the angle but this choice will soon be made a user option)

You can also enter pairs of parametric equations separated by semicolons. For example, entering "x=t^3-t; y=t^2" produces a simple loop graph. (The default domain is from t=-1 to 1, but you can change that by adding the domain limits after a second semicolon. So for example "x=t^3-t; y=t^2; -2,2" gives an extension of the loop.)

If you include letters other than 'x','y',or 'r' in the equation, they will be considered as parameters (unless occurring as parts of longer names or already defined as function names - see below), and controls to adjust their values will be presented in the 'Parameters' pane.
 
The 'Variables' tab is attached to a pane where you can define and control named variables

This pane has three parts whose sizes can be adjusted by sliding the dashed lines up and down.
(In the top section you can rename the 'Coordinates' but at present the axis labels don't properly reflect this, and the bottom section called 'Dependent Variables' is not yet functional, so it is the middle 'Parameters' section that is most useful)

The 'Parameters' pane is where you can define and control numerical constants.

If you include letters other than already defined  names in a formula, they will be considered as parameters.
Any parameters identified in an input formula will have a value controler added in the 'Parameters' pane. This consists of a text field where you can explicitly type in a numerical value, a slider for adjusting that value, and a pair of buttons labelled with "><" and "<>" for "zooming in" and "zooming out" (ie decreasing or increasing the range covered by the slider).

It is also possible to force the parser to recognize longer variable names by using the 'AddNewParam' button to add new parameters before they are used.
 
 

The 'Functions' tab pulls up a pane where you can define additional named functions.

In addition to recognizing the usual functions of elementary calculus, the parser will also recognize any function name defined in this window. Clicking on the 'NewFunction' button brings up a text field primed with the string "fn(x)=" (where n is the number of times the button has been clicked on). This can be completed or amended to produce any equation of the form "functionName(inputName)=expression" and on pressing Enter will define a function with the specified name.  (For example, entering "f(q)=q^2" will allow f(x) to replace x^2, and f(t) to replace t^2 in the graph definition examples.)
[Note: Unlike the original GraphExplorer, functions are not automatically graphed in this applet. Rather they become available for use in graph definitions. The 'Graphs' pane is where you can enter equations and formulas to be graphed.]
 

The 'Colour' pane is where you can colour a selected object.

Initially inactive, it becomes available when a graph's 'Color' button is clicked, or when a plotted object is clicked with the right mouse button with the 'alt' key down (ie 'alt-meta' mouse clicked on single button systems). The sliders adjust the mix of RGB in the color and the 'done' button shuts down the ColorSelecter. (It will pop back again when another color request is initiated).
 

Controlling the Region Displayed

The region controller used here is as in the original GraphExplorer (though the library does include other options). The mid-point coordinates and view dimensions can be adjusted by direct input or by using the zoom and pan buttons. It is also possible to use the mouse to move the grid by using a Shift-drag operation.
 

Mouse Actions in the Plot Pane

In addition to moving the region with a Shift-drag, the mouse can also be used to display labels and other objects.

Right clicking (or meta-clicking on single button systems) on an item produces an identifying label, and dragging with the right button displays a label with the current coordinates. (And if you want to get rid of it, left clicking on an existing lable makes it go away.)

The control key enables item creation. A control left click adds a plotted point to the display, and a control drag draws a line. Such items can be removed by a ShiftControl left click.

The alt key alters the state of an object. Alt-left click on a line toggles between segment and infinite display, and on a point near a graph 'fixes' it to the graph like a bead on a wire to constrain the effect of future drag operations. Alt-right click on a graph(or alt-meta click for  single button systems) brings up a color select panel.

Finally, clicking on a line with Control and Alt both pressed creates "run" and "rise" segments to complete the triangle.