GrafEq is an exceptional tool for exploring mathematics. As an introduction, here are some characteristics underlying the design of GrafEq:

• Mathematical
  Algebraic specifications are accepted in a broad variety of intuitive formats. For instance, a linear equation can be entered in the form y = mx + b, (y-k)/(x-l) = m, or ax + by + c = 0.
• Honest
  Any inaccuracy is explicitly revealed. For instance, although a pixel is very small, it is still an area rather than a point; therefore, the ‘coordinates’ of a pixel are reported using a range of values, such as (5.210±0.002, 17.001±0.002).
• Valid
  Graphs plotted are correctly. For instance, the graph of y<=sqrt(16-x²) differs from the graph of y>!sqrt(16-x²), because ‘<=’ is not equivalent to ‘>!’.
• Reliable
  No solutions will be missed: A pixel is off only if it contains no solutions at all; it stays on if it may possibly contain one or more solutions for the graph’s active relations. Extraneous solutions are eliminated when graphing is complete. The program can even use multiple colours to distinguish the pixels that have been proven to contain solutions, the pixels that have been proven to not contain solutions, and the pixels that are currently undecided.

  Hint: A closed and open version of a relation will often have identical graphs on the computer’s monitor. For example: the graph of x<5 will be identical to that of x<=5 unless the edge x=5 falls right between two adjacent pixels.

• Multi-faceted
  Each GrafEq graph can be presented in four forms:
  • Symbolic - an algebraic definition, typseset using mathematics notation, and presented in one or more algebraic relation windows. (Each relation can contain one or more constraints, and each constraint can be an equation, an inequality, a set description, or a conditional definition.)
  • Structural - a flow chart, or tree interpretation, presented in one or more structural relation windows. (Each relation is associated with an algebraic window and a structural window.)
  • Graphical - a cartesian or polar representation, presented in one or more view windows.
  • Printed - physical copies can be printed directly from GrafEq by means of its page editing window, or indirectly from other programs such as wordprocessors or drawing applications by first copying the relations and/or graph views from GrafEq, and then pasting into the other program.

Figure 1.1: A blank algebraic relation window

The algebraic relation window consists of the following elements:

• a title bar,
• a parameter bar, and
• some constraint fields.

The elements of the algebraic relation window are explained as follows:

Hint: An algebraic window’s title bar, parameter bar, and relation constraints are synchronized with, or equivalent to, their structural relation window counterparts. Custom ticks windows have the same setup for title and parameter bars. Therefore, the corresponding manual sections are equivalent.

Title bar

The title bar displays the graph title and the relation title followed by the “(Algebraic)” label, which helps identify the window as an algebraic relation window. The relation title can be changed using the parameter bar's title edit box. Figure 1.2 shows the title bar:

Figure 1.2: The title bar of an algebraic window

Parameter bar

The parameter bar is for customizing the relation title, starting and stopping graphing, modifying the graph colour, and changing the font size used to display contraints within the algebraic relation window. Figure 1.3 shows the parameter bar:

Figure 1.3: The parameter bar of an algebraic window

The elements, from left to right, are explained as follows:

Title edit box

The title edit box is for changing the relation’s title. Custom titles are useful in demonstrations and for print-outs. The default title, “Relation #1”, is easily changed: click on the title edit box to start editing. The new title will replace the old one in all menus, windows, and buttons that show the relation’s title.

Active checkbox

The Active checkbox is for starting and stopping relation plotting in open graph view windows. Clicking on the checkbox, or the checkbox name, toggles between the selected and unselected states. Deselecting the checkbox halts graph plotting and hides the relation from all graph views. Selecting the checkbox resumes graph plotting from where it left off and shows the relation in all graph views.

Hint: The Active checkbox is greyed out and unavailable if any relation constraints within the window are invalid.

Colour pop-up menu

The Colour pop-up menu is for changing the colour used for the relation in graph views. This control is useful for optimizing the clarity of graph views. The colour can be changed by clicking on the Colour box to access the pop-up colour menu, and then selecting the desired colour from the menu. For black-and-white graphs, a Pattern pop-up menu replaces the Colour pop-up menu. Colour computers can display black-and-white graphs. For more information, refer to the colour buddy section of the View Buddy Floating Window chapter.

Font Size pop-up menu

The Font Size pop-up menu is for changing the font size used to display relation constraints in the relation window. This control is useful for presentations via an overhead liquid crystal display (LCD) or a large screen television. The font size can be changed by clicking on the Font Size pop-up menu to bring up a menu of available font sizes, and then selecting a new size from the menu. All constraints of the current algebraic relation window will be displayed in the selected font size.

Hint: Some options in the algebraic relation window’s parameter bar do not change the behaviour or the appearance within the algebraic window itself. The Active checkbox affects the behaviour (graphing) of graph view windows. The Colour pop-up menu affects the appearance (graph colour) of graph view windows.

Constraint fields

Constraint fields are for entering relation constraints. The constraints entered are displayed in standard mathematics format. While a constraint is being entered, prompts appear at the bottom of the constraint field to offer basic tips during the specification process. Figure 1.4 shows a sample relation with two constraints (in two constraint fields):

Figure 1.4: An algebraic window with two constraint fields

While a constraint is being entered or changed,

• prompts appear at the bottom of its field,
• a blinking cursor is available,
• the Active checkbox of the algebraic window is unselected, and
• the easy button floating window is available.

Relation constraints are presented in standard mathematics format. When functions are entered using multiple characters, the characters are presented in red until GrafEq recognizes them as a function name: red signifies unrecognized entries. GrafEq can interpret “sin” as the sine function, instead of the product “s×i×n”, only after an argument, such as x, is entered. A good example is the square root function, when entered as “sqrt”: GrafEq will replace the letters with a graphical radical symbol only after an argument is started.

Standard mathematics format is desirable when including relations in handouts or documents, and can be done by copying relations from GrafEq and pasting them into a word processor. The formatted presentation also gives some visual feedback that constraints are interpreted as intended.

You can move from one constraint to the next by pressing the tab key; if there is no next constraint, a new one will be created. Pressing the tab key while the shift key is held down will move you on to the previous constraint; if there is no previous constraint, a new one will be created.

Hint: If the cursor is positioned away from the end of the relation string during editing, all characters to the right side of the cursor become unformatted.

Hint: With the default preference settings, unformatted characters are shown in solid red on colour displays, but are outlined in black (not filled in) on black-and-white displays.

Tip: The “Symbol” and “Times” TrueType fonts must be installed in order to produce attractive screen output, unless the font settings have been changed in the preferences.

Tip: More information on the format and structure of constraints and the parser is available in the How GrafEq Works chapter.

Conclusion

• Each algebraic relation window describes one relation; the relation can be defined using a single constraint or multiple constraints. Additional constraint fields can be brought up by pressing the tab key from within the algebraic relation window.
• Each graph can have one or more relations. Each relation is described in one algebraic window, and is normally assigned a colour different from that of the other relations. Additional algebraic windows can be opened by selecting New Relation from the Graph menu. Relations from the same graph are plotted on the same views, using the same coordinate system.
• New graphs can be opened by selecting New Graph from the File menu. When a new graph is opened, a new algebraic relation window appears automatically. The default graph title is “Graph #k”, where k is a new sequence number. Different graphs are unrelated; to view multiple relations in the same window, multiple relations are created within one graph.
• Besides the algebraic window, each relation may be presented using a tree structure in a structural relation window. Corresponding algebraic and structural windows are synchronized, so common attributes describing the relation can be changed in either window, and are reflected in both. Examples of common attributes include the relation title, graphing activity, and relation colour.
  The tree structure representation in the structural relation window is a new feature introduced in GrafEq 2.04. For details, see the Structural Relation Window chapter.
• The keystrokes field is a strip beneath the current constraint field. It shows the keystrokes used to produce the formatted relation in the constraint field. Keystroke information is especially useful when the intuitive mathematics symbols and notation have no obvious correspondance with the keys on a keyboard. Figure 1.5 shows a sample keystrokes field with its corresponding constraint field:
  
  Figure 1.5: An algebraic window with its keystroke field
  
  Prior to version 2.04, GrafEq always provided this information via the keystroke field. But as of GrafEq 2.04, keystroke information can be revealed within the constraint entry fields by positioning the cursor to the left of the entries. The keystroke field is hidden using the default preferences, although it is still available and remains useful for printing purposes. To display the keystrokes field, change the preferences.
• A relation entry can be completed by selecting the Active box of its algebraic window or by pressing the return key. When relation entries are completed, the algebraic window will collapse to display just the title bar, parameter bar, and constraint fields - prompts, the blinking cursor, the keystroke field, and the easy button floating window will disappear. The Active checkbox will be selected, but it can be freely deselected and selected without editing any constraints.
• Clicking within any constraint field of a completed algebraic window will return you to the edit mode and remove the relation from any opened graph views.

Buttons within the floating window are organized into groups, based on common usage: related symbols and operations are contained within each group. These groups are presented in alphabetical order. Figure 2.1 shows the easy button floating window displaying group headings, with only the Basic group of easy buttons expanded:

Figure 2.1: The default easy button floating window

Details about the easy buttons are explained in Appendix A, the Easy Buttons appendix. Following are some notes about the floating window:

• The easy button floating window appears when a relation or a custom ticks definition is being edited. It is otherwise hidden, but returns once a relation or a custom ticks definition is being edited.

  Hint: The easy button floating window disappears when relation editing is completed or a new window or program is activated.

• Each group of easy buttons can be expanded, to display its easy buttons below its title, or collapsed, to display just its title, by clicking on the name of the group, or the triangular arrow beside the name.

• Groups can be removed from, or added to, the easy button floating window by changing the preference settings, as follows:
  1. From the File menu, select Preferences;
  2. expand Algebraic Window;
  3. expand Easy Buttons;
  4. expand Access; then
  5. select the desired groups’ checkboxes under the Available column to add those groups; deselect the undesired groups’ checkboxes to remove those groups.

  Hint: If a group’s checkbox is not selected under the Available column, the buttons of that group will not be accessible at all. By default, the Measure, Numbers, and Roman groups are not available; all of the other groups are available. To have an available group automatically expand at startup, like the Basic group does by default, select the group’s checkbox in the Expand on Startup column. Expanding and collapsing available groups is often done, during a session, by clicking on the name of the corresponding group, or on the triangular arrow beside the name.

• To automatically keep the easy button floating window hidden while editing relations, change the preferences as follows:
  1. From the File menu, select Preferences;
  2. expand Algebraic Window;
  3. expand Easy Buttons; then
  4. deselect the Show easy buttons checkbox.

• To hide the easy button floating window during a session, click on the window’s close box, or deselect Easy Buttons from a Relation menu or a Custom Ticks menu.

Figure 3.1: A structural relation window displaying y=sinx

The structural relation window consists of the following elements:

• a title bar,
• a parameter bar,
• a tree structure field, and
• a tree control bar.

The elements of the structural relation window are explained as follows:

Hint: A structural relation window’s title bar, parameter bar, and its relation description are synchronized with, or equivalent to, their algebraic relation window counterparts. Custom ticks windows have the same setup for title and parameter bars. Therefore, the corresponding manual sections are equivalent.

Title bar

The title bar displays the graph title and the relation title followed by the “(Structural)” label, which helps identify the window as a structural relation window. The relation title can be changed using the parameter bar's title edit box. Figure 3.2 shows the title bar:

Figure 3.2: The title bar of a structural relation window

Parameter bar

The parameter bar is for customizing the relation title, starting and stopping graphing, modifying the relation colour, and changing the font size used to display elements of the structural representation. Figure 3.3 shows the parameter bar:

Figure 3.3: The parameter bar of a structural relation window

The elements, from left to right, are explained as follows:

Title edit box

The title edit box is for changing the relation’s title. Custom titles are useful in demonstrations and for print-outs. The default title, “Relation #1”, is easily changed: click on the title edit box to start editing. The new title will replace the old one in all menus, windows, and buttons that show the relation’s title.

Active checkbox

As with the Active checkbox in the algebraic window paramenter bar, the structural relation window’s Active checkbox is for starting and stopping relation plotting in open graph view windows. Clicking on the checkbox, or the checkbox name, toggles between the selected and unselected states. Deselecting the checkbox halts graph plotting and hides the relation from all graph views. Selecting the checkbox resumes graph plotting from where it left off and shows the relation in all graph views.

Hint: The Active checkbox is greyed out and unavailable if any relation constraints within the window are invalid.

Colour pop-up menu

The Colour pop-up menu is for changing the colour used for the relation in graph views. This control is useful for optimizing the clarity of graph views. The colour can be changed by clicking on the Colour box to access the pop-up colour menu, and then selecting the desired colour from the menu. For black-and-white graphs, a Pattern pop-up menu replaces the Colour pop-up menu. Colour computers can display black-and-white graphs. For more information, refer to the colour buddy section of the View Buddy Floating Window chapter.

Font Size pop-up menu

The Font Size pop-up menu is for changing the font size used in the tree nodes. This control is useful for presentations via an overhead liquid crystal display (LCD) or large screen television. The font size can be changed by clicking on the Font Size pop-up menu to bring up a list of available font sizes, and then selecting a new size from the menu.

Hint: The font size of the structural relation window affects the tree structure field, while the font size of the algebraic window affects the constraint fields. However, the title edit box, Active checkbox, and Colour pop-up menu of corresponding structural and algebraic relation windows are identical, and are synchronized. Changes in one will be reflected in the other. Some controls in the structural relation window’s parameter bar do not change the behaviour or the appearance within the structural window itself. The Active checkbox affects the behaviour (graphing) of graph view windows. The Colour pop-up menu affects the appearance (graph colour) of graph view windows.

Tree structure field

The tree structure field displays a relation using a tree structure. The tree structure is updated as the relation is entered. Figure 3.4 shows the sequence of tree structures displayed as the relation y=sinx is entered:

Figure 3.4: The sequence of tree structures displayed as y=sinx is entered

Tree control bar

The tree control bar is for customizing general attributes of the tree structure field, including node colour and tree structure format. Figure 3.5 shows the tree control bar:

Figure 3.5: The tree control bar of a structural relation window

The elements, from left to right, are explained as follows:

Node Colour pop-up menu

The node Colour pop-up menu is for changing the colour used to display tree nodes in the structural window. This control is useful for optimizing the clarity of the displayed tree structure. The colour can be changed by first clicking on the colour pop-up to bring up the pop-up colour menu, and then selecting the desired colour from the menu.

Combine Commutative checkbox

The Combine Commutative checkbox is for combining commutative operations within the tree structure. Clicking on the checkbox toggles between its selected and unselected states.

Figure 3.6 shows two tree structures for the same relation: The one the left, with the Combine Commutative unselected, shows a full, binary tree structure. The one on the right, with the Combine Commutative selected, shows a flatter tree structure, with commutative operations combined together.

Binary	n-ary

Figure 3.6: Alternative tree structure presentations

Conclusion

• Each structural relation window describes one relation; the relation can consist of a single constraint, or have multiple constraints. Constraints are linked together using an “and” operation.
• A structural window can be brought up by selecting Relation #n (Structure) from the Graph menu.
• Each structural relation window is associated with an algebraic window, and both describe the same relation. Corresponding structural and algebraic windows are synchronized, so common attributes describing the relation can be changed in either window, and are reflected in both. Common attributes include the relation title, graphing activity, and relation colour. For details about the algebraic window, see the Algebraic Relation Window chapter.
• A relation’s structural window, algebraic window, and easy button floating window can be shown or hidden by selecting or deselecting the appropriate option from a Relation menu. A Relation menu appears only when an algebraic or structural relation window is foremost.

The create view dialog box has two modes; a cartesian mode and a polar mode, as shown in figure 4.1, with cartesian on the left, and polar on the right:

Cartesian mode	Polar mode

Figure 4.1: Two create view dialog boxes

The create view dialog box contains the following elements:

• two graph radio buttons,
• a domain and a range pop-up menu,
• four bound edit boxes,
• a Cancel button, and
• a Create button.

Cartesian and Polar radio buttons

The Cartesian and Polar radio buttons are for changing the coordinate system used for plotting. They are in the upper left area of the create view dialog box. GrafEq sets the mode to Polar if r or q are present in the constraint specification; otherwise, GrafEq sets the mode to Cartesian. The mode can be changed by clicking on the other radio button or on its name.

Hint: If x and/or y, as well as r and/or q, is used in a relation specification, GrafEq sets the mode to Cartesian.

Domain and range pop-up menus

A variable pop-up menu is adjacent to each of the domain and range axes. They are for specifying the variables used for each axis of the new view. In the Cartesian mode, the variables default to x and y; in the Polar mode, they default to r and q. For a graph specified in variables other than x and y, or r and q, the variables can be changed by clicking on the respective pop-up menu to bring up the menu of available variables, and then selecting a new variable from the menu.

Bound edit boxes

Two sets of upper and lower bound edit boxes are adjacent to each of the domain and range variable axes. They are for specifying the bounds of the graph. In Cartesian mode, the default bounds are -10 to 10 for both x and y; in Polar mode, the default bounds are 0 to 10 for r, and 0 to 6.28 for q. Any of the bounds can be changed by first clicking inside their respective edit boxes, and then editing the values.

Hint: Instead of the mouse, the tab key or the arrow keys can be used to move among the edit boxes.

Create button

The Create button is in the center left area of the create view dialog box. It can be used in one of the following two ways:

• Simply click on the Create button and the following will happen:
  1. The create view dialog box will close.
  2. A view window, as specified in the create view dialog box, will open.
  3. Plotting of all active relations of the current graph will commence in the new view.
• While holding down the option key, click on the Create button and the following will happen:
  1. The create view dialog box will close.
  2. The view options dialog box, which provides additional options and controls, will appear. More details are described in the next section.

Cancel button

The Cancel button is for aborting the creation of a new graph view. It is in the bottom left area of the create view dialog box. Clicking on it will simply close the create view dialog box without creating a new view.

View Options Dialog Box

Figure 4.2: A view options dialog box

The view option dialog box contains the following elements:

• a Start Graphing Immediately checkbox,
• a Colour View checkbox,
• a View Size pop-up menu and arrow buttons,
• a Cancel button, and
• a Create button.

The elements of the view option dialog box are explained as follows:

Start Graphing Immediately checkbox

The Start Graphing Immediately checkbox is for controlling whether GrafEq starts plotting immediately after the graph view is created (by automatically selecting the Graph checkbox in the view window’s paramter bar), or waits until the Graph checkbox in the view window’s parameter bar is selected to start plotting. This option can be useful for presentation purposes. Clicking on the checkbox toggles between the selected and unselected states.

Colour View checkbox

The Colour View checkbox is for choosing between a colour view or a black-and-white view. The black and white option is useful for printing to a black-and-white printer. Clicking on the checkbox toggles between the selected and unselected states.

Hint: The black and white option is also useful for exporting to another application, such as a word processor, for further processing and subsequent black-and-white printing.

View Size pop-up menu and arrow buttons

The View Size pop-up menu and arrow buttons are for changing the size of the graph view on the monitor. GrafEq determines the default view size according to the size of the monitor used. The size can be changed in two ways:

• Click on the View Size pop-up to bring up a menu of available sizes, and then select a size from the menu.
• Click on the up arrow button to ascend (or the down arrow button to descend) through the list of available sizes.

Hint: View dimensions are measured in pixels. For high resolution display or printouts, select large values from the View Size pop-up menu to achieve best results.

Cancel button

This Cancel button is the same as the one in the create view dialog box. It is for aborting the creation of a new graph view. Clicking on it will simply close the view options dialog box without creating a new view.

Create button

The Create button is for setting off the following sequence of events:

1. The view options dialog box will close.
2. A new graph view, in accordance with the specifications given in the create view and view options dialog boxes, will be created.
3. If the Start Graphing Immediately checkbox was selected, plotting of all active relations of the current graph will commence; otherwise, GrafEq will wait until the Graph checkbox in the view window’s parameter bar is selected before plotting is started.

Conclusion

• Both the create view dialog box and the view options dialog box are modal; they must be finished with and closed before other GrafEq windows or features can be accessed. Pressing the Cancel button closes the box. Pressing the Create button removes the box and also creates a new view window, as explained earlier.
• If the create view dialog box is brought up by pressing the return key from an algebraic relation window, the relation specification is completed: the algebraic window collapses to display just the title bar, parameter bar and constraint field(s) - prompts, the blinking cursor, the keystrokes field, and the easy button floating window will all disappear.

A view window can be created by pressing the Create button of either the create view dialog box or the view option dialog box.

Hint: The create view dialog box appears automatically if a relation specification is completed by pressing the return key and the current graph does not have any open views. The dialog box can also be brought up by selecting New View from the Graph menu; additional views can be opened this way. The view options dialog box is accessed from the create view dialog box, by clicking on the Create button while holding down the option key.

Figure 5.1 shows a view window displaying a graph with one active relation, y=sinx:

Hint: Initial settings are adopted from both the create view and the view option dialog boxes.

Figure 5.1: A view window displaying a graph with one active relation, y=sinx

The view window consists of the following elements:

• a title bar,
• a parameter bar, and
• a view region.

The elements of the view window are explained as follows:

Title bar

The title bar displays the graph title and the view title. The view title can be changed using the parameter bar’s title edit box. Figure 5.2 shows the title bar:

Figure 5.2: The title bar of a view window

Parameter bar

The parameter bar is for customizing the view title, showing graphing status, controlling graphing activity, and displaying the percentage of the graphing area that has been proven to be correct. Figure 5.3 shows the parameter bar:

Figure 5.3: The parameter bar of a view window

The elements, from left to right, are explained as follows:

Title edit box

The title edit box is for changing the view’s title. Custom titles are useful in demonstrations and for print-outs. The default title, “View #1”, is easily changed: click on the title edit box to start editing. The new title will replace the old one in all menus, windows, and buttons that show the view’s title.

Graphing barber pole

The graphing barber pole relays graphing activity. It revolves while the graph is being plotted. It is replaced with a blank bar if graphing is stopped, but not completed. It disappears when graphing has been completed: an absent bar means that the displayed graph has been proven to be correct in every detail.

Graph (...) checkbox

The Graph (...) checkbox lets you pause, and resume, graphing. Graphing occurs when the checkbox is selected: clicking on the checkbox toggles between the selected and unselected states.

A progress report is shown in brackets; explanations of the common progress reports follow.

• “Waiting” means that graphing has not started.
• “>p% Proven”, with a single relation, means that more than p percent of the displayed pixels have been proven to be correctly coloured. With multiple relations, the average of the amounts proven for each relation is under-reported. For example, with two relations, one finished and the other just begun, the progress report would be “>49.9% Proven”.
• “Finished” means that graphing has completed.

The following explains the reports that are shown when graphing cannot continue.

• “Memory Stall” means that there is not enough available memory for graphing to continue. Increasing the amount of memory available to GrafEq may help.
• “Subdivisions Exhausted” means that even though the remaining (uncertain) pixels have been divided into the finest mesh GrafEq can use, no more proofs were forthcoming. Changing the syntactic form of the relation or zooming into the region of interest may help.
• “Precision Exhausted” means that GrafEq’s underlying number system does not have enough precision for graphing to progress. A correct graph of the desired area may be possible by introducing scaling and/or translation transformations into the relation.

Hint: If the Graph checkbox is deselected after graphing has finished, no change will occur.

View region

The view region is the main part of the view window: it displays the current view of the graph. Figure 5.4 shows a sequence of view windows displaying progressive stages of graph plotting, finishing with the completed plot.

Figure 5.4: Progressive stages of graph plotting

Hint: Ticked axes are applied to the view according to the current preference settings. They can be customized across sessions by modifying the preference settings, which are accessed by selecting Preferences under the File menu. The ticked axes can be turned off, for the current view window only, by deselecting Ticks in the colour view buddy. More controls over the ticked axes are available in the ticks view buddy. The ticks view buddy has many controls besides those for turning ticks on and off. Refer to the View Buddy Floating Window chapter for details.

Following is a summary of the controls of, and the various attributes of, the view window and its view region:

• The choice of the coordinate system used (cartesian or polar), the axes variables, and the initial view bounds (view scope) are made in the create view dialog box.
• A graph view’s on-screen dimensions are set in the view options dialog box.
• The choice between a colour graph and a black-and-white graph is made in the view options dialog box. This choice can be changed by using the colour view buddy.

  Hint: The colour view buddy has many controls besides simply turning colour display on and off. Refer to the View Buddy Floating Window chapter for details.

• Each view window has a view buddy floating window, which appears automatically when a graph window is foremost. The view buddy provides tools for interpreting and augmenting a graph and its views. Refer to the View Buddy Floating Window chapter for details.

Conclusion

• Each view window displays one view of a graph.

  Hint: A graph can include one or more relations; a relation can be described by one or more constraints.

  A graph can have multiple views. Each view shows all active relations of the graph within its specified view scope, using the options chosen for that view. Multiple views are useful for comparing different graphs, or for comparing different regions of the same graph.

• Additional views can be created by selecting New View from the Graph menu. The desired display may be obtained by customizing the options in the create view dialog box (and possibly the view options dialog box).

Figure 6.1: The view buddy floating window in its initial mode, colour

The view buddy has seven different modes. Colour mode is the initial mode, and is shown in figure 6.1 earlier. The other modes are accessable by clicking on the mode pop-up menu, and then selecting a new mode from the menu that pops up. The mode pop-up menu is at the very top of the view buddy floating window.

View buddy modes include:

• information mode,
• colour mode,
• ticks mode,
• zoom mode,
• draw mode,
• one point mode, and
• two point mode.

Before going into details about the different modes of the view buddy, following are some hints and general instructions about the floating window and its modes:

• The view buddy floating window appears only when a view window is foremost. It is otherwise hidden, but returns when a view window is activated.
• Options in the information, ticks, draw, one point, and two point modes apply only to the one view window directly associated with the view buddy. Options in the colour and zoom mode apply to all view windows of the associated graph.
• Following are some general instructions:
  • Selecting an option’s checkbox activates the option; clicking on a checkbox or its name toggles between its selected and unselected states;
  • Clicking on a pop-up menu opens the pop-up menu, so the desired option can be selected from it.
  • The colour mode is the initial mode since its options are useful for optimizing the clarity and readability of graph views.

Information Buddy

Figure 6.2: The information buddy displaying data for the graph view as a whole

The information buddy has several different sub-modes. “Totals” sub-mode is the default, as seen in figure 6.2 earlier. It displays data for the graph as a whole. Other modes display data for an individual relation of the graph. Each mode is accessed using the sub-mode pop-up menu, which is just below the mode pop-up menu at the top of the view buddy floating window

All information displayed applies to the immediately associated graph view only. The data elements, from top to bottom, are explained as follows:

Number of regions

This element shows the number of rectangular blocks remaining to be refined by GrafEq. When plotting is completed, the number of regions will be zero.

Memory allocated

This element shows the RAM being used by the graph. It varies with the complexity of the graph and the plotting technique employed. When plotting is completed, it shows the RAM required for displaying the plotted graph in the view window.

Scheduling pedigree

This element shows the plotting technique currently being used by GrafEq. Plotting techniques include the following:

• Dynamic: GrafEq is refining along the perimeter of the graph. This is the fastest technique, but requires the most memory.
• Perimeter scanning: GrafEq is refining along the perimeter of the graph. This is the second fastest plotting technique.
• Area scanning: GrafEq is refining everywhere inside the graph. This is the slowest plotting technique, and requires the least memory.
• Static: GrafEq is refining either by perimeter or area scanning.
• None: GrafEq has not refining the graph.

Evaluation speed

This element shows the up-to-the-moment average rate of graph processing. It varies with the complexity of the graph relations, processor speed, and memory resources. Evaluation speed is measured in Equation Evaluations Per Second (EEPS), Kilo EEPS (KEEPS), or Mega EEPS (MEEPS). When graphing is completed, this rate will be removed from the display.

Total number of evaluations

This element shows the total number of evaluations performed by the graphing process. It varies with the complexity of the graph view. The number of evaluations is measured in Equation Evaluations (EE), Kilo EE (KEE), or Mega EE (MEE). When graphing is completed, this number will stop changing.

CPU time/total time

This element shows the time spent on graph plotting. CPU time measures the amount of time the CPU has spent on graphing after factoring out the time spent on other tasks. Total time measures the total amount of time that has elapsed since graphing started. Any time that elapsed while the graphing process was deactivated (by deselecting the Graph (...) checkbox) is not included in the total time (or in the CPU time).

Tip: If CPU time is small relative to total time, closing non-essential windows and/or quitting other programs will increase graphing speed, as GrafEq will be able to use more CPU time for graphing.

Dimensionality

This element shows the dimension refining is occuring in. Even through GrafEq presents graphs using two dimensional views, GrafEq may be processing relations in higher dimensions if relations include variables other than the axes variables.

View size

This element is available only in the “totals” sub-mode. It shows the view region’s width and height, in pixels.

Show Work checkbox and colour pop-up menus

These elements are not available in the “totals” sub-mode: they are for showing detailed plotting information for a relation. When Show Work is selected, several colours will be used to convey graphing information, instead of the usual two, as follows:

• pixels that are proven to contain solutions have the appropriate relation’s colour;
• pixels that are proven to not contain solutions have the background’s colour; and
• pixels not yet proven to contain, or not to contain, solutions, have the work area colour.

Two colour pop-up menus are provided for the work area. If the colours selected are different, work areas will flash between the two selected colours. If they are the same, work areas will appear in the one colour, and will not flash. Different colours are useful when using a black-and-white display.

Background colour pop-up menu

This pop-up menu is for conveniently changing the colour used for the graph’s background. It might be useful for making work areas stand out.

Hint: The Background colour pop-up menu is also in the colour buddy (which is explained in the next section). The two Background pop-up menus are synchronized.

Colour Buddy

Figure 6.3: The colour buddy

All colour view buddy options apply to all view windows of the associated graph. The elements, from top to bottom in the colour buddy, are explained as follows:

Colour checkbox

The Colour checkbox is for toggling between a colour view and a black-and-white view. The black and white option is useful for printing to a black-and-white printer.

Blend checkbox

The Blend checkbox is for toggling between a blended graph and a layered graph. Blending is useful for showing intersecting areas when graphing multiple regions, such as those given by inequalities. A layered graph initially displays relations in the order that their respective relation windows are opened, earliest at the bottom, latest at the front. The layers can be reordered by hiding and showing relations. Click on a relation’s checkbox to hide and show that relation.

Background colour pop-up menu

The Background colour pop-up menu is for changing the colour used for the graph’s background. For black-and-white graphs, the background colour can be either solid black or solid white.

Ticks pop-up menu and checkbox

The Ticks colour pop-up menu is for changing the colour used for the ticks. For black-and-white graphs, the ticks can be either black or white. The Ticks checkbox is for showing and hiding the ticks.

(optional) Custom ticks colour pop-up menus and checkboxes

Each custom ticks pop-up menu and checkbox is similar to the Ticks pop-up menu and checkbox, but control display options for custom ticks instead of ticks.

Hint: Custom ticks colour pop-up menus are synchronized with the pop-up menus in the custom ticks windows.

(one or more) Relation colour pop-up menus and checkboxes

Relation colour pop-up menus are for changing the colours used to display relations. For black-and-white graphs, patterns are chosen instead of colours; a pattern pop-up menu is available instead of a colour pop-up menu.

Hint: Relation colour (or pattern) pop-up menus are synchronized with the Colour (or Pattern) pop-up menus in algebraic and structural relation windows.

The checkboxes adjacent to the colour pop-up menus are for showing and hiding relations.

Ticks Buddy

Figure 6.4: The ticks buddy presenting the preconfigured ticks

All options apply to the current view only.

The Show Ticks checkbox is the first option offered in the ticks buddy: it is just below the mode pop-up menu. When the checkbox is selected, ticks are shown.

Hint: By default preference settings, ticks are shown automatically when new views are opened. If GrafEq doesn't show ticks, graphing of complex relations might be faster. For details about changing preference settings, refer to the Preferences chapter.

The ticks buddy provides four different ticks sub-modes. “Preconfigured” is the default sub-mode, as seen in figure 6.4 earlier. Other sub-modes include “marks”, “outline”, and “parameters”. Each sub-mode is accessed using the pop-up menu below the Show Ticks checkbox.

The ticks sub-modes are explained as follows:

Preconfigured

The “preconfigured” sub-mode provides basic (simplified) tick options using four buttons. Each button sets the ticks to a preconfigured setting, as follows:

button 1: axes and sparse ticks; button 2: axes, sparse ticks, axes labels, and bounds; button 3: axes, dense ticks, axes labels, and bounds; button 4: axes, dense ticks, axes labels, and bounds, and grid cross ticks. Tip: This option can be used for making graph paper.

Figure 6.5: The “preconfigured” sub-mode of the ticks buddy

Marks, outline, and parameters

“Marks”, “outline”, and “parameters” sub-modes are for fine control of the appearance of the tick’s components. Option are offered using expandable lists.

• Collapsed lists can be expanded by clicking on the list’s title, or on the adjacent triangular arrow.
• Expanded lists reveal their elements, some of which may be expandable sub-lists. Revealed list elements can then be selected individually; expandable sub-lists can then be further expanded.
• Each list has a checkbox; all list items can be conveniently selected or deselected using this checkbox. A list which has elements that are selected as well as elements that are not selected displays a horizontal bar in its checkbox.
• Expanded lists can be collapsed by clicking on the list’s title, or on the adjacent triangular arrow.

Marks

This sub-mode is for selecting which tick components are shown. The various marks are useful for interpreting the graph, and for augmenting its display. Figure 6.6 shows all of the options:

Figure 6.6: The “marks” sub-mode of the ticks buddy

Outline

This sub-mode is for setting the thickness of the optional outline around tick components. A small area (an outline) may be placed around tick components, which can imporve readability, especially in cluttered graphs. Figure 6.7 shows the “outline” sub-mode:

Figure 6.7: The “outline” sub-mode of the ticks buddy

Parameters

This sub-mode is for changing the font size used for tick labels and the density of tick marks used. Figure 6.8 shows the “parameters” sub-mode:

Figure 6.8: The “parameters“ sub-mode of the ticks buddy

• Font Size pop-up menu is for changing the font size for any displayed variables and bound labels included in the ticks. The presence, or absence, of elements is governed by the “marks” sub-mode.
• Mark Density pop-up menu is for changing the density of ticks, dots, and crosses. The presence, or absence, of elements is governed by the “marks” sub-mode.

Zoom Buddy

Figure 6.9: A typical zoom buddy

All zoom buddy settings are shared between all view windows. The controls, from top to bottom in the zoom buddy, are explained as follows:

Colour pop-up menus

There is a pair of colour pop-up menus: the one on the upper left is for changing the colour used for the inside the zoom box; the one on the lower right is for changing the colour of used for the outline of the zoom box.

Zoom Out button

The Zoom Out button is for zooming out from the current graph view. The new view will have the same center, but both the domain and the range will be doubled; the new view will cover four times the area of the original view.

Keep Current View checkbox

The Keep Current View checkbox is for specifying whether the current view is kept or discarded after a new view is created by zooming in or out. Keeping the original view lets you compare graph views before and after zooming. Discarding the original view saves memory and keeps the machines’s desktop clean and tidy.

Zoom-in box

The zoom-in box specifies the scope of the new view that GrafEq will zoom into. It appears automatically when the mouse is within the graph view region. Zooming in is done by clicking the mouse, or by pressing the return key, while the area of interest is within the zoom-in box. The size of the zoom-in box, and thus the scope it specifies, can be changed by using the arrow keys: the left arrow narrows it, while the right arrow widens it; the up arrow heightens it, while the down arrow shortens it.

Hint: The mouse is usually at the center of the zoom-in box. When the mouse is still within the view region but far out towards one side, the zoom-in box stays within the current view region by sticking to an edge. As mentioned earlier, the colours of the zoom-in box are controlled by the colour pop-up menus at the top part of the zoom buddy.

Draw Buddy

Figure 6.10: A typical draw buddy

All drawings appear on the draw buddy’s immediately associated view only. The draw buddy provides access to a selection tool, an arrow tool, a box tool, a text tool, and a formula tool. Each draw tool is accessed by clicking on a button in the draw buddy window, and presents its options and controls using a separate floating window. If an object is currently selected, a tool’s controls will modify that object; otherwise, the controls will modify the default settings for that tool, which affect all objects subsequently created. Draw tools provide elaborate support for augmenting graph views. Details are available in Appendix III.

One Point Buddy

Figure 6.11: The one point buddy

One point buddy controls apply to its immediately associated view only. The buddy elements are explained as follows:

Coordinates field

The coordinates field displays the domain and range coordinates of the cursor point if the mouse cursor is within the graph view region. Coordinates values are given with error corrections.

Magnified view box

The magnified view box displays a magnified view of the region around the cursor.

Magnification pop-up menu

The Magnification pop-up menu is for changing the magnifying factor for the magnified view box. Supported magnification ranges from half-time(1/2x) to five-times(5x).

Two Point Buddy

Figure 6.12: The two point buddy

Two point buddy controls apply to its immediately associated view only. Two point buddy elements are explained as follows:

Coordinates field

The coordinates field displays the domain and range coordinates of A and B. Coordinates values are given with error corrections.

Attribute pop-up menu and attribute data field

The attribute pop-up menu is for changing the data type displayed in the attribute data field. Attribute options are listed and described as follows:

• trig - cosine, Dx, and sine, Dy of the line AB.
• slope - asymptote, h, and slope, m of the line AB.
• distance - distance, Dx, and Dy between A and B.
• degrees - angle formed by the line AB and a horizontal line through A, in degrees.
• radians - angle formed by the line AB and a horizontal line through A, in radians.

All attribute data are given with error corrections.

Figure 7.1: A blank custom ticks window

The custom ticks window consists of the following elements:

• a title bar,
• a parameter bar,
• a ticks control bar, and
• some tick specification fields.

The elements of the custom ticks window are explained as follows:

Title bar

The title bar displays the graph title and the custom ticks title. The default title, “Custom Ticks #1”, can be changed using the parameter bar’s title edit box. Figure 7.2 shows the title bar:

Figure 7.2: The title bar of the custom ticks window

Parameter bar

The parameter bar is for changing the title, showing and hiding the custom ticks, modifying the tick’s colour, and changing the font size used to display the tick’s defining relation. Figure 7.3 shows the parameter bar:

Figure 7.3: The parameter bar of the custom ticks window

The elements, from left to right on the custom ticks window parameter bar, are explained as follows:

Title edit box

The title edit box is for changing the title of the current ticks specification. Title editing is useful in demonstrations and for print-outs. The default title, “Custom Ticks #1”, can be changed by clicking on the title edit box to access the edit mode. The new title will be reflected in all GrafEq menus, windows, and buttons that show the custom tick’s title.

Active checkbox

The Active checkbox is for showing and hiding the custom ticks in all opened graph view windows. Clicking on the checkbox, or the checkbox name, toggles between the selected and unselected states. Deselecting the box hides the custom ticks from all graph views.

Hint: The Active checkbox is greyed out if the custom ticks definition is not valid yet.

Colour pop-up menu

The colour pop-up menu is for changing the colour of the custom ticks. The control is useful for optimizing the clarity of graph views. The default colour can be changed by clicking on the colour box to access the pop-up colour table, and then selecting the desired colour from the table.

Font Size pop-up menu

The Font Size pop-up menu is for changing the font size used for the tick’s definition in the custom ticks window. The control is useful for presentations via an overhead liquid crystal display (LCD) or large screen television. The default font size can be changed by clicking on the Font Size box to show the list of available font sizes, and then selecting from the list. All constraints used for the tick specification will be displayed in the selected font size.

Hint: Some options in the custom ticks window’s parameter bar do not change behaviour or appearance within the custom ticks window itself: The Active checkbox affects the appearance (custom ticks display) in graph view windows. The Colour pop-up menu affects the appearance (custom ticks colour) in graph view windows.

Ticks control bar

The ticks control bar is for changing the appearance of custom ticks in all view window(s) of the associated graph. The controls, useful in demonstrations and print-outs, are for optimizing the readability of the custom ticks. Figure 7.4 shows the ticks control bar.

Figure 7.4: The tick control bar of the custom ticks window

The elements, from left to right on the custom ticks window tick control bar, are explained as follows:

Labels pop-up menu

The labels pop-up menu is for changing which ends of the custom tick lines are labelled. This control does not affect polar views. Available options include:

• no labels,
• labels on the top (or left),
• labels on the bottom (or right),
• labels alternating between the top and the bottom (or left and right),
• labels alternating between the bottom and the top (or right and left), and
• labels on both the top and bottom (or left and right).

Figure 7.5 shows example views with various label options:

No labels


Labels on the top


Labels on the bottom


Labels alternating between the top and the bottom


Tick labels alternating between the bottom and the top


Labels on both the bottom and the top

Figure 7.5: Example views with various label options

Line pop-up menu

The line pop-up menu is for changing the style used to display the custom tick lines. This control affect both polar and cartesian views. Available options include:

• no lines,
• dotted lines,
• dashed lines, and
• solid lines.

Figure 7.6 shows views with various line options:

No lines


Dotted lines


Dashed lines


Solid lines

Figure 7.6: Views with various line options

Symbolic simplification pop-up menu

The symbolic simplification pop-up menu is for changing how much simplification is to be applied to the custom tick labels. This control does not affect polar views. Available options include:

• none, for no simplication;
• bracketed, for simplified presentation of any parts with brckets; and
• complete, for a simplified presentation.

Figure 7.7 shows views with various symbolic simplification options:

No symbolic simplification


Bracketed symbolic simplification


Complete symbolic simplification

Figure 7.7: Views with various symbolic simplification options

Label font size pop-up menu

The label font size pop-up menu is for changing the font size of the tick line labels. This control does not affect polar views. Eight sizes are available, ranging from very small fonts to very large fonts.

Outline pop-up menu

The outline pop-up menu is for changing the thickness of any outlining around displayed custom tick lines and labels. Outlining is useful when parts of the custom ticks are obscured by graph regions. This control does not affect polar views. Available options include:

• no outline,
• minimal outline,
• moderate outline, and
• maximal outline.

Figure 7.8 shows views with various outline options:

No outline


Minimal outline


Moderate outline


Maximal outline

Figure 7.8: Views with various outline options

Display variable pop-up menu

The display variable pop-up menu is for selecting whether or not the variable is displayed within tick labels. This control does not affect polar views. figure 7.9 shows views with, and without, the variable being displayed:

Variable displayed in labels


Variable not displayed in labels

Figure 7.9: Views with, and without, the variable being displayed

Hint: All options in the custom ticks control bar only affect the appearance of custom ticks in graph view windows. They do not affect the behaviour or appearance within the custom ticks window itself.

Custom tick constraint fields

Custom tick constraint fields are for specifying the custom ticks. They also display the entered specification in standard mathematics notations. Prompts appear at the bottom of the field, while a constraint is being entered, to offer basic tips during the specification process. Figure 7.10 shows examples simple and complex custom tick specifications:

Simple tick specification	Complex tick specification

Figure 7.10: Custom ticks windows displaying simple and complex specifications

Custom ticks constraints should be specified in terms of the domain or range variable. The simple tick specification in figure 7.10 creates vertical ticks in multiples of p, which is especially useful for graphing trigonometric relations. Additional samples of useful tick specifications are provided at the end of this manual chapter.

While a constraint is being entered or changed,

• prompts appear at the bottom of its field,
• a blinking cursor is available,
• the Active checkbox of the custom ticks window is unselected, and
• the easy button floating window is available.

Custom ticks constraints are presented in standard mathematics format. When functions are entered using multiple characters, the characters are presented in red until the GrafEq recognizes them as a function name. Red signifies unrecognized entries, and is used here because GrafEq can interpret “sin” as the sine function instead of the product “s×i×n” only after an argument, such as x, is entered. The function “sqrt” is a good example that demonstrates GrafEq formatting functions as they are recognized.

Standard mathematics format is desirable for inclusion in handouts or documents (by copying from GrafEq and pasting into a word processor). The presentation also gives some visual feedback that constraints are interpreted as intended.

Additional constraint fields can be brought up by pressing the tab key.

Hint: If the cursor is positioned away from the end of the relation string during editing, all characters to the right side of the cursor are unformatted.

Hint: With default preference settings, unformatted characters are red in colour displays, and outlined in black-and-white displays.

Tip: The Symbol and Times TrueType fonts must be installed in order to produce attractive screen output, unless default preferences are changed.

Tip: More information on the format and structure of constraints and the parser is available in the How GrafEq Works chapter.

Following are some additional hints and tips about custom ticks and the custom ticks window:

• Custom ticks are usually either domain or range interpretation aids: For cartesian views, they would be vertical or horizontal lines. For polar views, they would be radial lines eminating from the origin, or concentric circles about the origin. Custom ticks are usually specified in terms of the domain variable or the range variable. For both cartesian and polar views, custom ticks can be specified using r or q.
• Each custom ticks window describes one set of custom ticks; the custom ticks can be defined using a single constraints, or by using multiple constraints. Additional constraint fields can be brought up by pressing the tab key from within the custom ticks window.
• Each graph can have one or more sets of custom ticks. Each set is described in its own custom ticks window. Additional custom ticks windows can be opened by selecting New Custom Ticks... from the Graph menu.
• The keystrokes field is a strip beneath the current constraint field. It shows the keystrokes used to produce the formatted relation in the constraint field. Keystrokes information is especially useful when the intuitive mathematics symbols and notation do not correspond to general keyboard entries. Figure 7.11 shows a sample keystrokes field with its corresponding constraint field:
  
  Figure 7.11: A custom ticks window with its keystroke field
  
  Prior to version 2.04, GrafEq always provided this information via the keystroke field. But as of GrafEq 2.04, keystroke information can be revealed within the constraint entry fields by positioning the cursor to the left of the entries. The keystroke field is hidden using the default preferences, but remains useful for printing purposes. To display the keystroke field, change the default preference setting.
• A custom tick entry can be completed by selecting the Active box of its custom ticks window or by pressing return. When custom tick entries are completed, the custom tick window will collapse to display just the title bar, parameter bar, ticks control bar, and constraint fields - prompts, the blinking cursor, the keystroke field, and the easy button floating window will disappear. The Active checkbox will be selected, but it can be freely deselected and selected without editing any constraints.
• Clicking within any constraint field of a completed custom ticks window will return you to the edit mode and remove the corresponding set of custom ticks from any opened graph views.

Conclusion

• For a single vertical tick line through (p,3), enter x=p.
• For horizontal tick lines through multiples of 1/2p, enter y=k(p/2).
• For “log-paper”, enter x=10^k.
• For “cross-hairs”, enter x=p for tick #1, and y=logp for tick #2.
• For circular ticks, in cartesian views, enter r=k.
• For radial ticks, in polar views, enter q=k(p/6).

  Hint: Single tick lines or cross-hairs are desirable, especially if memory is low or only specific regions are of interest. Instead of using custom ticks, adding constraints such as x=10k; 0<k<4 allows GrafEq to concentrate on the region of interest and not bother with custom ticks.

Figure 8.1: A typical page window

The page window consists of the following elements:

• a title bar,
• a parameter bar, and
• a page region.

The elements of the page window are explained as follows:

Title bar

The title bar displays the graph title and the page title. The default page title can be changed using the parameter bar's title edit box. Figure 8.2 shows the title bar:

Figure 8.2: The title bar of a page window

Parameter bar

The parameter bar is for customizing the page title and changing the scaling factor used to display the page on the screen. Figure 8.3 shows the parameter bar:

Figure 8.3: The parameter bar of a page window

Following are the elements, from left to right, on the parameter bar:

Title edit box

The title edit box is for changing the title of the current page. Title editing is useful during demonstrations and for print-outs. The default title, “Page #1”, can be changed by first clicking on the title edit box to enter the edit mode, and then changing the title name. The new title will be reflected in all menus, windows, and buttons that show the page title.

Page scale pop-up menu

The page scale pop-up menu is for changing the scaling factor used to display the page on the screen. As a default, GrafEq uses the largest scaling factor that can display an entire page without scrolling. The scale can be changed by clicking on the page scale pop-up menu to bring up a list of available scale factors, and then selecting from the list. A small scaling factor can be used for display on a small monitor; A large scaling factor can be used to magnify the page content for close examination and precise editing.

Page region

The page region displays the page being prepared for printing. The licensee name is always displayed on the top left corner of the page. The page can also include graphs, formulae, text, pointer arrows, and rectangular boxes. When a page window is foremost, a page buddy floating window appears to offer drawing tools, for augmenting the page, and printing tools, for printer control and printing.

Hint: The drawing tools of the page buddy are the same as those offered by the view buddy.

Conclusion

• Each page window is associated with, and displays the contents of, one graph only. The graph can have multiple views, any of which may be included in a page.
• To print separate graphs, separate page windows must be opened.
• Additional page windows may be opened by selecting New Page from the Graph menu.

Figure 9.1: The page buddy floating window

The page buddy provides the following general drawing tools:

• a selection tool, for selecting objects;
• an arrow tool, for drawing arrows and lines;
• a box tool, for drawing rectangular boxes;
• a text tool, for adding text annotations; and
• a formula tool, for adding formula annotations.

• a view tool, for adding views; and
• a printing tool, for selecting, configuring, and using a printer.

• The page buddy floating window appears only when a page window is foremost. It is otherwise hidden, but returns when a view window is activated.
• Each tool is accessed by clicking on a button in the page buddy floating window. Controls and options for the selected tool are presented in an associated, but separate, draw tool floating window.
• The selection tool does not have any options. Therefore, it does not bring up the draw tool floating window until an object is selected.
• Following are some general instructions:
  • Selecting an option’s checkbox activates the option; clicking on a checkbox or its name toggles between its selected and unselected states;
  • Clicking on a pop-up menu opens the pop-up menu, so the desired option can be selected from it.
  • The colour mode is the initial mode since its options are useful for optimizing the clarity and readability of graph views.

General Draw Tools

Graph Tools

Figure 9.2: The Graph Draw Tool Floating Window

The elements, from top to bottom in the graph draw tool floating window, are explained as follows:

View pop-up menu

The view pop-up menu is for selecting a view to the clipboard, for subsequent pasting to the page.

Enforce Minimum Size checkbox

The Enforce Minimum Size checkbox is for toggling between imposing or not imposing a 25% limit for a pasted graph views’s minumum size. If the option is not selected, pasted views can be shrunk arbitrarily small.

Print Tools

Figure 9.3: Printer controls in a draw tool floating window

The elements, from top left across to bottom right in the print draw tool floating window, are explained as follows:

Page Setup button

The Page Setup button is for accessing the operating system’s standard page setup window. Options for the printer, paper size, source and orientation, graphic resolution, and number of copies to be printed can then be selected.

Print button

The Print button is for accessing the operating system standard print control window. The print control window is for sending the current print page to a printer, while it also supports the same options as offered in the page setup window.

Show Printer Page checkbox

The Show Printer Page checkbox is for toggling the display of the limits that the print head can reach on the physical paper.

Page size and printer resolution

The page size displays the print page dimensions in inches; the resolution displays the resolution in dots-per-inch, “dpi”.

Reset Guides button

The Reset Guides button is for changing all margins back to the default setting. Default margins are set to a minimum, and the guides would overlap with the printer page.

Show Guides checkbox

The Show Guides checkbox is for toggling the outlining, with dotted line, of the page area on which page contents are to be printed.

Pop-up menus for Left, Right, Top and Bottom Guides

The guide pop-up menus are for changing the respective print margins.

Hint: Margins can also be set by clicking at the respective guide box on the page, then dragging it to the desired position.

• File menu,
• Edit menu,
• Graph menu,
• Relation menu, and
• Custom Ticks menu.

Hint: File and Edit menus are always present; but Graph menu is present only when there is one or more active graphs; Relation menu is present only when an algebraic window or structural window is foremost; and Custom Ticks menu is present only when a custom ticks window is foremost.

File Menu

Figure 10.1: The File menu

• Opening and creating
  • New Graph creates a new graph, and brings up a new algebraic window for the new graph’s first relation. The new graph will be assigned a new title, using a new sequence number.
  • Open... lets you open a saved graph from disk. Preference files may also be opened, although a few preferences (such as interface language) only apply at startup.

• Configuring
  • Preferences brings up the preferences window, which is for setting up the GrafEq environment for current and/or future sessions. Many elaborate controls are accessible from the File Preferences menu; for details, refer to the Preferences chapter.
  • Register... lets you register your copy of GrafEq. This menu option is not present in physically licensed copies of GrafEq.

• Closing and saving
  • Close Graph closes all windows associated with the foremost graph.
  • Close Window closes the foremost window.
  • Save Graph stores the current graph or view to disk. GrafEq supports three file formats, as explained following:
    • Full Graph - all information, of the chosen graph, including relations, views, pages, ticks, and the current plotting state is saved. A full graph can later be opened in the exact same state as when it was saved. Partially plotted graphs can be refined from where it was left off. Full graphs can use a significant amount of disk space.
    • Skeleton Graph - all information, of the chosen graph, excluding the current plotting state, but including relations, views, pages, and ticks of the foremost graph is saved. When a skeleton graph is re-opened, graph plotting will start over, because the skeleton file format does not save the plotting state. Skeleton graphs use very little disk space.
    • Simple View - only the chosen graph view is saved. With Windows, it is saved using the WMF format. With a Macintosh computer, it is saved using the PICT format. Simple view files can only be re-opened in GrafEq as a picture. However, the may also be opened, and modified, in painting and drawing programs.
  • Save Preferences saves the current preferences.
  • Save Graph As... lets you save a graph or view to a new file. Other than letting you choose a new name and location for the file, Save As... is identical to Save.
  • Save Preferences As... lets you save the preferences to a new file. GrafEq looks for a preference file named “GrafEq Preferences” at startup, which is the default name used for saved preferences.

• Printing
  • Page Setup... opens the operating system’s standard page setup window. Options may include the paper size, paper source and orientation, printign resolution, and the number of copies to be printed.
  • Print... opens the operating system’s standard print window. The print window is used for sending a page to an available printer. The print window may also include some options that influence the printnig process.

  Hint: GrafEq supports printing via the creation of pages which can be invoked by selecting New Page from the Graph menu. For details, refer to the Page Window chapter.

• Selecting
  Selection controls provide access to all open graphs. Each is listed using its current title. Selecting a graph causes it to becomes the current graph, and bring its first algebraic relation window to the front.

• Terminating
  • Halt Graphing stops graph plotting in all open views.
  • Quit closes the current GrafEq session.

Edit Menu

• Cut removes the selected content from its current location, and keeps a copy on the clipboard.
• Copy copies of the selected content to the clipboard.
• Paste inserts the clipboard’s contents into the current position. If some content is selected (and there is no current position), the selection is replaced with the clipboard’s contents.

• When the view window is foremost:
  • the Cut and Paste operators are greyed out (unavailable), because the view window does not allow changes to it; the view window displays graph views and the plotting process.
  • the Copy operation is supported, because the contents of the view region can be copied, and then pasted to other applications.
• When the page window is foremost, the entire Edit menu is greyed out (unavailable), editing in the page window is provided via the page buddy floating window.
• When the structural relation window is foremost, the entire Edit menu is greyed out (unavailable), because the structural window does not allow changes to it; the structural window displays the tree structure of its associated relations.

Graph Menu

Creating elements

Creation controls are for creating new relations, views, custom ticks, and pages.

• New Relation creates an algebraic relation window for entering a new relation.
• New View... opens the create view dialog box, from which a graph view can be created.
• New Custom Ticks creates a custom ticks window for specifying a new set of custom ticks.
• New Page creates a page window for preparing a page for printing.

Displaying elements

Display controls provide access to all open relations, views, custom ticks, and pages; each listed with its respective title. The window for the selected item becomes the foremost window.

Relation Menu

Custom Ticks Menu

Figure 11.1: The GrafEq preferences window

GrafEq preferences are organized mainly by windows; other general preference for file access control and interface languages are also provided.

Before going into details about the preferences, following are some general hints about the effects preference settings and some general use instructions:

• Most preference settings take effect immediately; some take full effect only in subsequent sessions, but full preference changes can be activated immediately by quitting and restarting GrafEq.
  
• Default means that the settings will apply to all appropriate cases, and across sessions, every time GrafEq is started. These settings can be changed ‘on the fly’, while GrafEq is running, but the changes will apply only to individual graphs or windows, and will not affect the defaults. The next time GrafEq is started, default settings would be back in effect.
• Selecting an option’s checkbox activates the option; clicking on a checkbox or the checkbox’s name toggles between its selected and unselected states;
• Clicking on a pop-up menu brings up its pop-up list or table, so the desired option can be selected from it.
• Many preference controls are offered in lists. Lists are expandable, as explained following:
  • Lists can be expanded by clicking on the triangular arrow adjacent to, or the name of, the respective list.
  • Expanded lists reveal the respective list elements and/or sub-lists. List elements revealed can then be selected individually; and sub-lists can then be further expanded. The same list rules apply to sub-lists.
  • Some lists also come with checkboxes adjacent to their respective triangular arrows. The list box can be selected to conveniently select all elements of the list.
    Fully selected lists have check marks in their checkboxes; partially selected lists have horizontal bars in their checkboxes; fully unselected lists have blank checkboxes.

Windows Preferences

Algebraic Window Preferences

Figure 11.2: Algebraic window preferences

• Algebraic Window - Clipboard
  Clipboard Preferences are for setting appearances and formats to be used when exporting from GrafEq’s algebraic windows to the clipboard. Figure 11.3 shows clipboard preferences:
  
  Figure 11.3: Algebraic window clipboard preferences
  
  • Add padding space around equation checkbox
    Select the option if it enhances layout appearance when relations are pasted to a wordprocessing document.
  • Export to clipboard as... format pop-up menu
    Select some text to export a relation in simple ASCII characters only; select a picture to export it exactly as it appears in the algebraic window; and select some text and a picture to include both text and picture information.
  • Use ... bitmap in place of a picture checkbox and resolution pop-up menu box
    Select bitmap and select low resolutions to save memory and disk space; select high resolutions for smoother and clearer bitmaps.
  • Include formatting details in text clip checkbox
  
  
• Algebraic Window - Easy Buttons
  Easy buttons preferences are for setting defaults to show, or not show, the easy button floating window; and setting access to each easy button group. Figure 11.4 shows easy buttons preferences:
  
  Figure 11.4: Algebraic window easy buttons preferences
  
  • Show easy buttons checkbox
  • Access
    • Expand on Startup checkbox and Available checkbox for each easy button group. Select Available to display the heading of the respective easy button group; select Expand on Startup to display the heading as well as individual easy buttons of the respective easy button group. By factory default, the Roman characters and numbers group are unavailable because they are already available from the standard keyboard. Refer to to Appendix A, the Easy Buttons appendix for details of easy buttons.
    
    
• Algebraic Window - Formatted Display
  Formatted display preferences are for setting font type, default font size, and appearances of entries in the algebraic window’s constraint field(s). Figure 11.5 shows formatted display preferences:
  
  Figure 11.5: Algebraic window formatted display preferences
  
  • Defaults:
    • On startup, warn user if proper fonts are not available checkbox
    • Precise character layout checkbox
    • Default font size... pop-up menu, with up and down arrow buttons
      The up arrow button is for stepwise size increment, and the down arrow button is for stepwise size decrement.
  • Fonts
    • Effects sample - This area displays the effects of the current font options set under this manual section.
      The highlighted character(s) represents the currently selected character kind. The appearance of these highlighted characters and the font settings currently displayed correspond with each other. The kind of characters highlighted is synchronized with the selection displayed in the character kind pop-up menu, and can be changed by either clicking within the effects sample box, or from the character kind pop-up menu. The character kind pop-up menu is described next.
    • Character Kind pop-up menu - The pop-up menu lists formatted character kinds (entries to the left side of the mouse cursor) first, then a separator line, and finally unformatted character kinds (entries to the left side of the mouse cursor).
    • Font setting controls include:
      • a Font pop-up menu,
      • a Bold checkbox,
      • a Italic checkbox,
      • a Outline checkbox, and
      • a Colour pop-up menu.
      Any font setting changes made will apply only to the currently selected character kind, as highlighted in the effects sample box, and indicated in the character kind pop-up menu.
    
    

    Hint: The highlighted character kind in the sample box and the current option in the kind pop-up menu are synchronized. The format and appearances of the highlighted characters are synchronized with font type, boldface, italic and outline type, and colour settings selected/displayed. Any changes in the font options takes immediate effect in the sample box. Symbol character kinds, for both formatted and unformatted sections, should use symbol font only. Experiment with the various options and see the effects.

  
  
• Algebraic Window - Keystroke Display
  

  Hint: As of GrafEq version 2.04, keystroke information is available directly from the constraint entry fields, by positioning the cursor to the left of the entries. The keystroke field remains useful for printing purposes, but has been set to be hidden by factory default.

  Keystroke display preferences is for setting to show, or not show the keystroke field, and setting font type, and appearances of entries in the keystroke field. Figure 11.6 shows keystroke display preferences:
  
  Figure 11.6: Algebraic window keystroke display preferences
  
  • Show keystroke line checkbox
  • Fonts These fonts preferences are essentially the same as that for the formatted display, they apply to entries to the algebraic window’s keystroke display field.

• Algebraic Window - Teleprompter
  The teleprompter preference is for toggling to show, or not show, teleprompter (which is the user hint shown at the bottom of an active constraint or custom tick specification field). There is just one preference option, which is provided by the Show teleprompter checkbox. Figure 11.7 shows the teleprompter preference:
  
  Figure 11.7: Algebraic window teleprompter preferences
  

Memory Monitor Window Preferences

Figure 11.8: Memory monitor window preferences

• Show memory monitor on startup checkbox
  Deselect this option if memory is scarce. The memory monitor can always be brought up instantly while GrafEq is running, by selecting Memory Monitor from the help menu.
• Displayed Data options list:
  • Labels checkbox
  • Numbers checkbox - Exact quantities of memory, in Bytes, KBytes, or MBytes can be displayed, in addition to percentages for data items.
  • Show largest available block checkbox
  • Amalgamate used memory checkbox
  
  
• Window appearance options list:
  • Resizable window checkbox
  • Horizontal window checkbox - If unselected, vertical window would be used.
  • Show bars checkbox - If selected, proportional bars will be used to present data items, in addition to percentages, and numbers.
  • Update data every... time interval pop-up menu

Page Window Preferences

Figure 11.9: Page window preferences

• Updates preference is provided by the Update from view window every... time interval pop-up menu.
  
  
• Decoration preference is provided by the default font size pop-up menu.

Preferences Window Preferences

Figure 11.10: Preferences window preferences

• Need password to access preferences checkbox
• Need password to save preferences checkbox

  Hint: The same password will be used for controlling access to preference settings and saving, as well as access to file loading and saving. A user specific password is supplied with registration.

Structural Window Preference

Figure 11.11: Structural window preference

Title Window Preference

Figure 11.12: Title window preferences

View Window Preferences

Figure 11.13: View window preferences

• View Window - Clipboard
  Clipboard preference is for setting format and resolution to be used when exporting from GrafEq’s view windows to the clipboard. Figure 11.14 shows the clipboard preference:
  
  Figure 11.14: View window clipboard preference
  

  Clipboard preference is provided by the Use ... bitmap in place of a picture checkbox and resolution pop-up menu. Select bitmap and select low resolutions to save memory and disk space; select high resolutions for smoother and clearer bitmaps.

  Hint: Relations and graph views are always exported in screen resolution, bitmap resolution selected affects only ticks, and text and elements added in with drawing tools.

• View Window - Colours
  Colours preferences are for setting default appearances, and colours and patterns for graph views. Figure 11.15 shows colour preferences:
  
  
  Figure 11.15: View window colours preferences
  
  • Display views in colour, if possible checkbox - If unselected on a colour system, GrafEq will display views in patterns.
  • Blend layers checkbox - If unselected, colours or patterns will pile up in opaque layers.
  • Colours list:
    • Background colour pop-up menu
    • Layer colours pop-up menus - The first 15 layers of default colours can be specified.
  • Patterns list:
    • Background pattern pop-up menu
    • Layer patterns pop-up menus - The first 15 layers of default patterns can be specified.

• View Window - Information
  Information preferences are for setting updating frequency, and optional display data items for the view window’s information view buddy. Figure 11.16 shows information preferences:
  
  
  Figure 11.16: View window information preferences
  
  • Updates preference is for setting the time interval at which the display data selected are to be updated in the view window’s information view buddy. The preference is provided by the Update info every... time interval pop-up menu.
    
    
  • Display Data preferences list:
    • Region count checkbox
    • Memory usage checkbox
    • Scheduling pedigree checkbox
    • Equation Evaluation Rate checkbox
    • Total Equation Evaluations checkbox
    • CPU time checkbox
    • Real time checkbox
    • Dimensionality checkbox
    • Show work areas checkbox
    • Work area colours colour pop-up menus
    • Pixel size checkbox
    Refer to the View Buddy Floating Window chapter for detailed meaning and useful application of information display data.

• View Window - Ticks
  Ticks preferences are for setting when to show created ticks, and setting optional default components and default appearances of ticks in graph views. Figure 11.17 shows information preferences:
  
  Figure 11.17: View window ticks preferences
  
  • Creation preference is for setting when to show newly created ticks in graph views. The preference is provided by the After creating a new view, display ticks... time pop-up menu, and
    the or... state pop-up menu. Displaying immediately might be convenient; displaying later might speed up graphing.
    
    
  • Defaults preferences are for setting optional default components and default appearances of ticks in graph views. It starts with an attribute pop-up menu, from which marks, outline, or parameter displays can be set.

    Before going into details about these ticks preferences, following are some hints on the use of the tick preferences controls:

    • For expandable list menus, select the list checkbox to select all elements of the list, or expand lists (by clicking on the triangular arrow adjacent to the list name or the checkbox) to select individual elements, such as lines, ticks and axes, or domain and range, etc...
    • Deeper lists, such as the arrowheads and bounds lists, can be expanded further to deeper levels. For the case of arrowheads and bounds lists, each of domain and range lists can be further expanded to reveal the respecitve min and max elements, so each element can be selected individually.

    Defaults preferences are listed as follows:

    • Marks default components preferences lists:
      • Axes checkbox, and list which is expandable to the following:
        • Lines checkbox, and list which is expandable to Domain and Range checkboxes.
        • Ticks checkbox, and list which is expandable to Domain and Range checkboxes.
        • Arrowheads checkbox, and list which is expandable to Domain and Range checkboxes, and lists of which each is further expandable to Min and Max checkboxes.
      • Labels checkbox, and list which is expandable to the following:
        • Variables checkbox, and list which is expandable to Domain and Range checkboxes.
        • Bounds checkbox, and list which is expandable to Domain and Range checkboxes, and lists of which each is further expandable to Min and Max checkboxes.
      • Grids checkbox, and list which is expandable to Dots and Crosses checkboxes.
      
      
    • Outline preferences are for setting defaults to outline optional tick marks, and by what thickness. Options are listed as follows:
      • Outline... thickness pop-up menu
      • Axes checkbox, and list which is expandable to Lines, Ticks and Arrowheads checkboxes.
      • Labels checkbox, and list which is expandable to Variables and Bounds checkboxes.
      • Grids checkbox, and list which is expandable to Dots and Crosses checkboxes.
      
      
    • Parameter preferences are for setting default fault size and mark density, naturally by the Font Size and Mark Density pop-up menus,
  
  
• View Window - View
  View preferences are for setting
  
  Figure 11.18: View window view preferences
  
  • Size preference is for setting default view size, naturally by the Default view size... size pop-up menu.
  • Calculation preferences are for setting rounding and real number representation rules, as listed following:
    • Perform arithmetic with outward rounding checkbox - If unselected, inward rounding would be used.
    • Use extra wide relations and intervals checkbox
  • Refining preferences are for setting triggers to the graphing process and real number handling methods, as listed following:
    • Start graphing automatically checkbox - If unselected, GrafEq would wait for user command to start.
    • Perturb bounds checkbox - If selected, graphed lines will have uniform thickness. Every new view created has its bounds enlarged by about 0.01% of the view size, so that lines like x=0 will fall upon a column or row of pixels rather than in between two columns or rows.
  
  
• View Window - Zoom
  The Zoom preference is for setting defaults to keep or replace the original view on zooming, naturally by the Keep the old view after zooming checkbox. Figure 11.19 shows information preferences:
  
  Figure 11.19: Algebraic window teleprompter preferences
  

Files Preferences

Language Preference

Conclusion

GrafEq also has many features of interest to the sophisticated user, including its relation parser, plotting technique, and memory management.

Hint: Useful and interesting hints and tips relevant to the immediate topic are offered throughout this GrafEq manual. Most chapters are also concluded with general hints and tips for the chapter topic.

The Algebraic Relation Parser

• A relation can contain one or more constraints. Each constraint must be a valid relation. The graph of a relation given by multiple constraints is the intersection of the graphs given by the constraints taken individually. This interpretation can be further understood by opening a structural relation window.
• A relation can be an equation, an inequality, a set description, or a conditional or piecewise definition.
• A relation can be explicit or implicit:
  • explicit if one of its variables is isolated on one side of the relation definition;
  • implicit if neither of its variables is isolated.
• Conditional or piecewise descriptions are entered as follows: y<{ x if x²+y²>=5, sinx if (-1<x<2p), -5 if x ∈ {8, 18, 28, 38} }
• Relations may be given using set notation, as follows: a<{b,c}, which is interpreted as the union of a<b and a<c. GrafEq does not use a type-system, so that x={1,2} is the same as x∈{1,2}.
• Relation definitions must be in a valid form: for example; braces, brackets, and parenthesies must be properly closed.

Tip: Older versions of GrafEq perform little symbolic manipulation, and plotting is primarily numeric. Natural pre-processing of expressions such as replacing “x3 - x3” with a simple “0” might enhance efficiency significantly. Starting with version 2.04, GrafEq does simple pre-processing, so that it will recognize (and perform) simple replacements that improve graphing speed without changing the graph of the relation. Rewriting and simplifying relations must be done very carefully! For example, “>=!” should not be re-written as “<” because they are not equivalent in all cases. Figure 12.1 shows the graphs of “y >=! Arccosx” and “y < Arccosx” as plotted by GrafEq; they are not equal because Arccosx is undefined for x>1 and x<-1: Graph of y >=! Arccosx Graph of y < Arccosx Figure 12.1: Graphs plotted by GrafEq shows that “>=!” and “<” are not the same

The Plotting Technique - Successive Refinement

1. A view region is divided into rectangular sub-regions.
2. Each sub-region is examined:
   • if GrafEq can prove that the sub-region does not contain any solutions that satisfy any of the active relations, the sub-region does not form part of the final graph, and it is removed from the graph;
   • if GrafEq can prove that all points within the entire sub-region does satisfy an active relation, the entire sub-region forms part of the final graph, and is never looked at again;
   • if the sub-region lies within a single pixel and GrafEq can prove that a point within the sub-region does satisfy an active relation, the sub-region forms part of the final graph, and is never looked at again;
   • if GrafEq cannot prove one of the above three propositions, the sub-region is further examined:
     • if the sub-region cannot be further subdivided
       • because it has become so small that the current number system cannot represent the required numbers, then GrafEq will state that “precision is exhausted”,
       • because it has become too small relative to the size of a pixel, then GrafEq will state that “subdivisions are exhausted”;
     • otherwise, the sub-region will be further subdivided, and each smaller sub-region will be examined just like its predecessor, as explained above.

   Hint: A graph is drawn using the following rules: pixels that form part of the final graph have the (possibly blended) colours of the relevant relations; pixels that do not form part of the final graph have the colour of the view region’s background; and pixels that are not yet determined have the (possibly blended) colours of the relevant relations, or if the Show Work option, in the information view buddy, is selected, the colour of the work area.

Hint: GrafEq can correctly plot irregular graphs such as singularities and narrowly spiked curves. Figure 12.2 shows three stages in the plotting of “x(x+1)2+y(y+1)2=0”: Figure 12.2: GrafEq plotting a relation with a singularity, in successive refinement

Hint: GrafEq displays a “signature” at point-discontinuities. Consider the relation y=(x-1)(x+1)/(x-1). Its graph can be anticipated to be the line y=x+1 less the point (1,2). However, when the relation is fed to a computer, while graphers using common sampling plotting techniques would likely miss the issue of the point discontinuity altogether, even GrafEq with its robust graphing engine cannot omit any pixels for the point (1,2), because the pixel containing (1,2) also contain many solutions in the real number continuum that satisfy the relation. In such cases, rather than just graphing the solid line y=x+1, GrafEq will often make the abnormaly apparent, by displaying a signature at the point-discontinuity, as shown in figure 12.3: Figure 12.3: Graph with point discontinuity signature at x=1 Starting with version 2.04, GrafEq has an improved graphing engine which may not display discontinuity signatures (older versions may not either; the signature is a side-effect of the graphing process). Future versions will have added features, which will allow users to see undefined portions of a graph.

Memory and Speed

• Quit any screen savers and applications that are opened but currently non-essential.
• Close any windows that are currently non-essential.
• Turn off the “Keep this view after zooming” option in the Zoom View Buddy.
• Turn off graph ticks wherever possible and appropriate.
• Set view windows and view scopes as small as possible.
• Keep the number of active plotting views to a minimum, just one if possible.
• If plotting additional relations in a view window, set other relations to inactive.
• Set all colours or patterns to solid black wherever appropriate.

• the memory being used by each view and each relation;
• the time spent exclusively for graphing versus the total time elapsed, because the computer also spends time in re-drawing windows, and monitoring external events such as mouse clicks, key presses.

• preprocess the relation, such as replacing “x²-x*x” with “0”;
• use same colour output instead of black and white patterns;
• use solid black instead of patterns or colours during plotting - patterns and colours can be reset after plotting is completed.

Further Reading

Operations, functions and symbols provided by the easy buttons require certain formats, and take varying number of arguments. Most entries are mathematically intuitive. Relation editing windows also provide hints at the bottom of the window view to prompt for the next step of action.

Algebra Easy Buttons

Power Functions	Inverse Power Functions

Figure A.1: Algebra Easy Buttons

The entities x and y within the easy buttons represent expressions that should be entered after the button-click; and the entity a represents arguments that should be entered prior to the button-click. Each button is explained as follows:

• For the a², and a³ buttons, enter an argument first; then click the button;

• For the 10^x, and e^x buttons, click the button first; then enter the power expression (represented by x). The effects can be observed in the relation visual format, the relation tree structure, and the relation graph.
  For example, to enter “e^33.54”,
  1. click the e^x button;
  2. enter the power expression represented by x, which is “33.54” in this example.
  Hint: Parentheses would be necessary for power terms such as x/y, or 2xy, but not if the power is a single simple term.
  Experiment with GrafEq to see the differences when these power terms are entered with, and without parentheses.

• For the a^x button, enter the argument first; then click the button; then enter the power index expression (represented by x).
  For example, to enter “x²+y³=5^2xy”,
  1. enter “x”;
  2. click the a² button;
  3. enter “+y”;
  4. click the a³ button;
  5. enter “=5”;
  6. click the a^x button; and
  7. enter the expression “(2xy)”.
  Hint: The same parenthesis rules apply to a^x, 10^x, and e^x, which means that parentheses are necessary for power terms such as x/y, or 2xy, but not for single simple terms. Experiment!

• For the square-root, log, and ln buttons, click the button first; then enter an argument, which can be a single simple term or a complex term enclosed in appropriate parenthesis.

• For the cube-root button, click the button first; then enter the argument and finish the term with a closing parenthesis.

• For the xth-rooty button, click the button first; then enter the root index expression (represented by x) follow by a comma; then enter an argument expression (represented by y); and finally finish the term with a closing parenthesis.
  For example, to enter “5th root of 2x+y”,
  1. click the xth-rooty button;
  2. enter the root index expression represented by x, which is “5” in this example,
  3. follow by a comma;
  4. enter the argument expression represented by y, which is “2x+y” in this example,
  5. finish term with a closing parenthesis.

• For the log_xy button, click the button first; then enter an expression for the log base (represented by x); followed by the argument expression (represented by y).
  For example, to enter “log₂x”,
  1. click the log_xy button;
  2. enter the logarithm base represented by x, which is “2” in this example;
  3. enter the argument represented by y, which is “x” in this example.
  Hint: Parentheses are not required if the base and the argument are single simple terms or fractions of single simple terms, such as x, y, x/y, 2, 3, or 2/3. Parentheses are necessary for terms such as x+y, 2xy, or for two consecutive numeric terms that would otherwise be treated as one number. Try it out!!

Arithmetic Easy Buttons

Figure A.2: Arithmetic Easy Buttons

The *, -, and / are all infix binary operators (used in between two arguments), while +, -, ± can be either infix binary, or prefix unary (used in front of one argument, which can be a single term or a complex term enclosed in appropriate parentheses).

Basic Easy Buttons

Figure A.3: Basic Easy Buttons

Factoring Easy Buttons

Figure A.4: Factoring Easy Buttons

• For mod, lcm, gcd, and gamma, click the button, then enter arguments.
  Gamma takes one argument, which can be a single term or a complex term enclosed in appropriate parentheses. Mod takes two arguments while lcm and gcd takes two or more, each separated by a comma, and the last one followed by a closing parenthesis.
• For !, enter an argument first, then click the button. The single argument can be a single term or a complex term enclosed in appropriate parentheses.

Greek Easy Buttons

Set a	Set j	Set s
Set A	Set J	Set S

Figure A.5: Greek Easy Buttons - Lower and Uppercase Characters

Integer Easy Buttons

Figure A.6: Integer Easy Buttons

• For both floor and ceiling operations, click the opening button, then enter the argument, then click the closing button. The single argument can be a single term or a complex term enclosed in appropriate parentheses.
• For round, and trunc, click the button first, then enter an arguments.
  The single argument can be a single term or a complex term enclosed in appropriate parentheses.

Measure Easy Buttons

Figure A.7: Measure Easy Buttons

• For |x|, click the button first, then enter an arguments, which can be a single term or a complex term enclosed in appropriate parentheses, then close by clicking the button again.
• For signum, click the button first, then enter an arguments, which can be a single term or a complex term enclosed in appropriate parentheses.
• For angle, click the button first, then enter an arguments, which can be a single term or a complex term enclosed in appropriate parentheses.

Numbers Easy Buttons

Figure A.8: Numbers Easy Buttons

Numbers can be conveniently entered via the computer keyboard; this easy button set is provided for completeness, and is turned off in preferences settings by factory default.

Order Easy Buttons

Figure A.9: Order Easy Buttons

• Click the ordering buttons first, then enter arguments.
  
• Min and max return the minimum and maximum respectively, of specified arguments, and min_x and max_x return the xth smallest and the xth largest respectively, of specified arguments.
• Min and max each takes two or more arguments, each separated by a comma, and the last one followed by a closing parenthesis.
• For min_x and max_x, enter a positive integer base first, then enter an appropriate number of arguments, each separated by a comma, and the last one followed by a closing parenthesis.
  Hint: The number of argument should be equal to or larger than the value of x.

Relational Easy Buttons

Direct Relations	Negated Relations

Figure A.10: Relational Easy Buttons

All relational symbols are infix binary operators(used between two arguments).

Roman Easy Buttons

Set a	Set j	Set s
Set A	Set J	Set S

Figure A.11: Roman Easy Buttons - Lower and Uppercase Characters

Roman characters can be conveniently entered via the computer keyboard; this easy button set is provided for completeness, and is turned off in preferences settings by factory default.

Set Easy Buttons

Figure A.12: Set Easy Buttons

• For braces, brackets, and parentheses, click the opening button, then enter argument(s), then click the closing button to finish.
• Pairs of braces, brackets, and parentheses can all be used to control precedence.
• Pairs of braces can also be used to enclose set elements (with a comma separating every two successive set elements).
• “If” is used within braces for conditional definition, with one or more sets of a value term followed by if, then the condition. “If” should be preceded a space, and followed by a space. A typical example is y={x if x>0, 0 if x=0, -x if x<0}
• “is member” and “is not member” are for set membership definition.
• “,” is required for separating elements or arguments.

Trig Easy Buttons

Circle Functions	Hyperbola Functions	Square Functions	Diamond Functions
Inverse	Inverse	Inverse	Inverse
Circle Functions	Hyperbola Functions	Square Functions	Diamond Functions
Inverse	Inverse	Inverse	Inverse
Circle Relations	Hyperbola Relations	Square Relations	Diamond Relations

Figure A.13: Trig Easy Buttons

All trig operators are prefix binary, to be used in front of one argument which can be a single simple term or a complex term enclosed in appropriate parenthesis.

To conclude this section of the manual, following are some additional hints and tips about the easy button floating window:

• The display of each group of easy buttons in the floating window can be toggled on or off by clicking the triangular arrow adjacent to the corresponding set name.
• Easy button groups altogether with the respective headings can be hidden from the easy button floating window, across sessions, by unselecting the respective “Available” checkboxes from the File Preferences 's Windows - Algebraic Window - Easy Buttons - Access menu. By factory default, all but Measure, Numbers, and Roman buttons are selected, but a smaller selection may be appropriate for a beginners’ class or a special lesson.
• All functions, operations and symbols may also be entered via the keyboard. The key-combinations for special characters and operators are listed in Appendix E, the Keyboard Shortcuts appendix.

Algebraic Functions

• Exponentiation includes a², a³, and the complete range of simple and complex power functions. Each takes one base term, and one power term.

  Hint: Besides easy buttons, entry of exponentiation can also be initiated by pressing the up arrow key or the caret key.

• Radical includes sqrt, root(3,x), and the complete range of simple and complex root functions. Each takes one root index term, and one radical term.
• Logarithm includes log (for base 10), lg (for base 2), ln (for natural logarithm or base e), and the complete range of simple and complex logarithm functions. Each takes one base term, and argument term.

  Hint: Besides easy buttons, entry of logarithm base can also be initiated by pressing the down arrow key.

Arithmetic Operations

• Addition (+), Subtraction (-), and Add-and/or-Subtract (±) can be either infix binary with the operator in between two argument terms, or prefix unary with the operator in front of one argument term. Hint: Besides the easy button, the ± symbol can be entered with the control-+ key combination.
• Multiplication (*) and Division (/ or ÷) are infix binary. Hint: Besides the easy button, the ÷ symbol can be entered with the control-/ key combination.

Factoring Functions

• Modulo (mod) takes two argument terms, and returns the positive remainder of the division of the first argument term by the second argument term.
• Least Common Multiple (lcm) and Greatest Common Divisor (gcd) each takes two or more argument terms, and are generalized to real valued arguments.
• Gamma (gamma) function is a continuous extension of the factorial function. For any non-negative integer x, gamma(x+1) is equal to x!.
• Factorial (!) is postfix unary with the operator following one argument term. It is a recursively defined function: for any non-negative integer x, (x+1)! returns (x+1)x!, and 0! returns 1.

Integer Functions

• Floor (⌊x⌋) returns the closest integer smaller than or equal to the argument value.

  Hint: Besides the easy buttons, the floor symbols can be entered with the control-[ and control-] key combinations.

• Ceiling (⌈x⌉) returns the closest integer larger than or equal to the argument value.

  Hint: Besides the easy buttons, the ceiling symbols can be entered with the control-{ and control-} key combinations.

• Round (Round) and round (round) returns the rounded or closest interger from the argument value. Round (with a capital ‘R’) rounds up, and round (with a lowercase ‘r’) rounds down; which means that if the argument value ends with ‘.5’, such as 4.5, Round would return 5 and round would return 4.
• Trunc (trunc) returns the integer portion of the argument value, by deleting the fractional part. It is the same as the floor function for positive argument values, the same as the ceiling function for negative argument values.

Measure Functions/Magnitude Operators

• Absolute Value (|x|) takes one argument, and returns the value without the sign.
• Signum (signum) takes one argument, and returns -1 for a negative argument, 1 for a positive argument, and 0 for an argument that equals 0.
• Angle (angle) takes two arguments x and y, and return the angle between the point (x,y) and the x-axis.

Order Functions

• Minimum (min) and Maximum (max) each takes two or more arguments, and returns the smallest, and largest value respectively.
• xth minimum (min_x) and xth maximum (max_x) each takes a number of argument equal to or larger than the value of x, where x is a positive integer. The xth smallest, and xth largest value respectively would be returned. Hint: If x does not evaluate to an integer between 1 and the number of arguments, both functions would be undefined.

Relational Functions

Set Functions

• Set element listings are to be enclosed within pairs of braces ({}), with a comma (,) separating every two successive set elements.
• Set membership can be set with is member (e) and is not member (e). Hint: Besides easy button, is member of can also be entered with the control-= key combination.
• Conditional definitions are described within pairs of braces ({}), with one or more sets of a value term followed by if (if), then the condition. An example is: y={x if x>=0; -x if x<0}

Trig Functions and Relations

• Trig Functions include circular, hyperbolic, square, and diamond functions.
  • Circular trig functions are based on the circle, and include sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
  • Hyperbolic trig functions are based on the hyperbola, and include the six functions of the circular ones, but with an appended “h”. They are sinh, cosh, tanh, csch, sech, and coth.
  • Square trig functions are based on the square, and include the six functions of the circular ones, but with an appended “q”. They are sinq, cosq, tanq, cscq, secq, and cotq.
  • Diamond trig functions are based on the diamond parallelogram, and include the six functions of the circular ones, but with an appended “d”. They are sind, cosd, tand, cscd, secd, and cotd.
• Inverse Trig Functions also include circular, hyperbolic, square, and diamond functions. Each collection has the same trig functions, but with a preceding “Arc”. They include Arcsin, Arccosd, and so on.
• Inverse Trig Relations also include circular, hyperbolic, square, and diamond relations. Each collection has the same trig operations, but with a preceding “arc”. They include arcsin, arccosd, and so on.

To conclude this section of the manual, following are some additional hints and tips about GrafEq functions, relations, and operations:

• Each argument or term can be a simple term or a complex term that can contain any number of functions, operations, and values. Complex terms might need to be enclosed in appropriate/necessary parentheses.
• Degree (°), Minute ('), and Second ('') returns p/180, p/(180*60), and p/(180*60*60) respectively. Hint: 5'33'' is interpreted as the multiplication product of 5p/(180*60) and 33p/(180*60*60).
• Precedence of operations:
  1. Bracket-like operators, such as brackets, parenthesis, braces, floor, ceiling, absolute value, and conditional/piecewise definition
  2. Postfix operators, such as factorial
  3. Exponentiation
  4. Multiplication and division
  5. Prefix operators, such as negation
  6. Addition and subtraction
  7. Comparisons, such as equal-to, less-then, and greater-then
  • Precedence for function application is not clear cut. For example, sin2x is parsed as sin(2x); multiplication has higher precedence than the sine function. However, sinxcosx is parsed as (sinx)(cosx); multiplication has higher precedence in sin2x, but has lower precedence in sinxcosx.
  • The visual mathematics formating GrafEq’s algebraic window presents often reflect whether a specification is entered correctly. If in any doubt, parentheses or brackets can always be used.
    Tip: The structural tree structured relation description is a new feature introduced in GrafEq 2.04, and can be used for accurate confirmation.

• a selection tool, for selecting objects;
• an arrow tool, for drawing arrows and lines;
• a box tool, for drawing rectangular boxes;
• a text tool, for adding text annotations; and
• a formula tool, for adding formula annotations.

Each draw tool is accessed by clicking on a button in the draw buddy window, and presents its options and controls using a separate floating window. If an object is currently selected, a tool’s controls will modify that object; otherwise, the controls will modify the default settings for that tool, which affect all objects subsequently created.

The Selection Tool

• At most one object can be selected at a time.
• A new object is selected by first ensuring you have the selection tool as the current tool, and then clicking on the desired object. The cursor will change into a hand with a pointing finger when you are using the selection tool, and you are over an unselected object.
• The selected object can be edited using the tools provided in the draw tools floating window
• The draw tool floating window associated with the type of object selected automatically comes up to provide you with tools for editing. For an explanation of a drawing tool’s controls, refer to the manual pages for that tool.
• With the selection tool as the current mode, a selected object can be deselected by:
  • selecting another object,
  • pressing the return key, or
  • clicking on an area where there are no objects to select.
• If no object is selected, no drawing tools will be displayed.

• An object is automatically selected after it is created.
• Any drawing object can be selected with the selection tool by clicking on the object.
• Control points are found in all selected objects. They characterize the selected object, and denote its limits.
  • Unselected objects do not display any control points.
  • For formulae, text, and boxes, control points denote the object’s four corner points and four edge midpoints.
  • For arrows, control points denote the arrow’s starting (source) and ending (destination) points.
• Control points are used for resizing and re-orienting objects.
  • For formulae, text, and boxes, the corner points control the two adjacent edges, while the midpoints control just the one immediate edge.
  • For arrows, control points are for changing the length, and endpoints, of the arrow.
  • To use a control point: first place the cursor directly above a control point, so that the cursor changes into a small triangle pointer, and then drag the control point to its new position.
• The selected object can be the top object, or in a layer underneath other objects.
  • The selected object can be brought up to the topmost layer by pressing the up arrow key once. The selected object can be pushed down to the bottom layer by pressing the down arrow key once. Fitting into specific layers in between the topmost and the bottom layer is not supported in current versions. But you can bring one object to the top, and then bring other objects on top of it, to relayer objects as desired.
• The selected object can be moved by first placing the cursor over the object, but away from any control points or other objects, and then dragging the object to a new location. When dragging is possible, the cursor will change into an open (grabbing) hand.
• The selected object can be deleted by pressing the del key or the delete key once.

Arrow Tools

To resize, re-orient, move, or delete an arrow, refer to the selection tool manual section. Line or arrow drawing can be customized using the arrow tools, displayed and explained as follows:

Figure C.1: Arrow tools floating window

• Source radio buttons are on the left; destination radio buttons on the right. Choose an appearance for the source point using a source button; choose an appearance for the destination point using a destination button.
• Arrowhead Size pop-up menu - click on this pop-up menu to open the arrowhead size menu to select a desired size. Different sizes have the same proportion between width and length. The selected size applies to both source and destination ends.
• Arrowhead Width pop-up menu - click on this pop-up menu to access the width pop-up menu, to select a desired width. Different width have the same length. The selected width applies to both source and destination ends.
• Line Thickness pop-up menu - click on this pop-up menu to access the thickness popup list, to select a desired thickness. The selected thickness applies to the source, the line, as well as the destination.

Formula Tools

Figure C.2: Formula Tools Floating Window

• Font size pop-up menu and finetune arrows - each click on the finetune up arrow increases the font size by 1; each click on the finetune down arrow reduces it by 1.
• Transparent background checkbox
• Use bitmap checkbox
• Resolution pop-up menu

Text Tool

Figure C.3: A Text Editing Window

• The text region is for entering and displaying the text.
• The Cancel button cancels adding text, and closes the text editing window.
• The Ok button completes text editing, closes the text editing window, and adds the text to the graph view or page.

To resize, move, or delete a text object, refer to the pointer control and active objects manual sections. Text entry and editing can be formatted using text tools, displayed and explained as follows:

Figure C.4: Text Tools Floating Window

• Font pop-up menu
• Font Size pop-up menu and finetune arrows each click on the finetune up arrow increases the font size by 1; each click on the finetune down arrow reduces it by 1.
• Text format checkboxes for Bold, Italic, and Outline fonts
• Text alignment radio buttons for Left, Center, and Right alignment
• Edit Text button - click on this edit button to return to the text editing window, to change the text contents. This button is greyed out (inactive) when the text tool is first accessed and a text object has not been entered yet.
• Transparent background checkbox

Box Tools

Figure C.5: Box Tools Floating Window

• Box background radio buttons for Transparent background radio button, Colour background radio button and colour pop-up menu, and Pattern background radio button and pattern pop-up menu.
• Line Thickness pop-up menu - The line thickness affects the border of the box.

Numbers

Numbers can be entered via the computer keyboard, the numeric keypad, or the easy button floating window.
Tips: Infinity (oo) is a constant, and can be entered by presing control-8.

Roman Letters

Roman letters can be entered via the computer keyboard, or the easy button floating window.

Greek Letters

Greek characters can be entered via the easy button floating window, or using the keyboard key combinations listed in figure D.1 following:

Sets

• discrete, containing elements listed within a pair of braces, or
• continuous - continuous sets can contain
  • open ranges described within a pair of parentheses, or
  • closed ranges described within a pair of square brackets.

Conclusion

• GrafEq relations can be specified in a wide latitude:
  • Virtually all alphabetical characters are acceptable as variables.
  • At the Create View window, any alphabetical characters entered in a relation, as well as x, y, r, and q can be specified as domain and range variables.
  • For cartesian (rectangular) graphs, domain (horizontal axis) and range (vertical axis) variables default to x and y; for polar graphs, q and r. All defaults can be modified.
  • Hint: In both cartesian and polar graphs, r defaults to the distance from the origin, and q to the position angle in radians.
• Some characters are reserved in GrafEq to be values, therefore, they cannot be used to represent variables. These values include: p, e, oo, as well as all numeric digits.
• Variable names must be atomic; xy means two variables, and cannot be used as one variable name.

File Menu

Edit Menu

View Buddy

To conclude this section of the manual, following are some additional hints and tips about GrafEq’s keyboard shortcuts:

• Either uppercase or lowercase letters can be used.

• For a general description for the GrafEq program, visit the online description page at http://www.peda.com/grafeq/.
• For an overview of the program’s functionalities, visit the online specifications page at http://www.peda.com/grafeq/.
• For a complete reference, consult the GrafEq manual.

To launch the program

1. Double-click on the GrafEq icon, as seen in figure I.1 following:
   
   Figure I.1: The GrafEq icon
   

2. A splash screen appears, which gradually reveals the GrafEq title screen, as seen in figure I.2 following. Press the space bar once to clarify the splash screen, and once more to remove it.
   
   Figure I.2: The GrafEq logo
   

3. The Relation #1 algebraic window then opens automatically, as seen in figure I.3 following:
   
   Figure I.3: An Algebraic Relation Window
   

To enter a relation

1. Ensure that the caps lock key is not depressed.
   
   
2. Follow the step sequence, which will be provided, to enter the relation as seen in figure I.4 following:
   
   Figure I.4: y=sqrt(x)sin1/x
   

   The necessary step sequence is as follows (Refer to the hints immediately following regarding easy buttons and prompts.):

   1. Type in “y=”;
   2. Click the sqrt() easy button;
   3. Type in “x”, then “)”, noting the prompt at the bottom of the relation window, that says “Press ) to finish the term”;
   4. Type in “sin1/x”.
   Some general hints about entering relation specifications:
   • Use the keyboard to enter letters, digits, brackets, and simple symbols such as +, -, *, /, !, >, <, =, ...
   • Use easy buttons to enter special characters and operators, such as algebraic powers and roots. Figure I.5 following shows the easy button floating window:
     
     Figure I.5: Easy Button Floating Window
     

   • The relation window also displays a prompt message to guide the user. In an empty relation window, as in figure I.5, the help message prompts the user to enter a relation. Other appropriate messages appear at different stages of relation input.
   • The relation window parses entries as a relation specification is being entered, and displays them in standard mathematics format. The sequence of keyboard keys pressed can be revealed by moving the cursor from the end of the entry line to the left.
   • To make changes to the relation, or correct typing mistakes:
     1. position the cursor, with the mouse or the left and right arrow keys, to where changes are to be made; (Any relation portions now behind the cursor would become unformatted, as a sequence of keyboard entries; don’t worry about them, they will become formatted again when relation entry is completed, or when the cursor is placed back to the end of the entry line.)
     2. press the delete key to erase the character to the left of the cursor, and
        press the del key to erase the character to the right of the cursor;
   
   
3. The prompt message at the bottom of the window now states “Press return to graph...” Press return now. The Create View window will then appear. Figure I.6 following is the create view window:
   
   Figure I.6: The Create View Window
   

To specify a viewport

1. GrafEq can plot both Cartesian and Polar graphs, and will automatically set the default mode according to the relation entered. The graph mode can be changed by clicking its radial butto. Since we have entered Relation #1 as a rectangular relation, Cartesian mode is active (by default) in our Create View window, and we will not change that.
   
   
2. We will not change the default domain (horizontal) and range (vertical) bounds, but any of them can be changed by:
   1. clicking on the box of the bound to be changed; then
   2. editing the number in the box; and
   3. repeating the steps for any other bounds to be changed.

To create a graph

1. Create the specified view by clicking the Create button, or simply pressing return as the Create button is the default button. (The thick border around the Create button denotes that it is the default button.)
   The graph view, with an accompanying view buddy floating window appear as in figure I.7 following:
   
   Figure I.7: Graph view of y=sqrt(x)sin1/x, and the view buddy
   

To apply or remove axes and scales

1. Go to the view buddy, click and hold down the mouse on the mode pop-up menu on the top of the floating window; a pull down list appears.
   
   
2. Drag the mouse down the list to Ticks, then release the mouse to enter Ticks mode.
   
   
3. Click on each of the four buttons in turn to see the different ticks. Figure I.7 uses the second button.
   
   
4. The Show Ticks checkbox on top of the buttons is selected. To hide the ticks, click on the checkbox once to toggle ticks inactive. (One more click and the ticks will be toggled active, into the last state selected.)
   
   

To alter the viewport region by zooming

1. Return to the view buddy, and enter zoom mode by selecting Zoom from the mode pop-up menu which is on the top of the floating window.
   
   
2. The Zoom-Out button is on the view buddy now, but there is not a button for zooming in, since we have to specify the region to be zoomed into.
   
   
3. Place the mouse cursor within the graph view; a yellow zoom box will appear.
   
   
4. Place the zoom box at the graph’s interesting portion, close to the center, as seen in figure I.8 following:
   
   Figure I.8: Zooming into a graph's interesting area, with the zoom buddy
   

5. Click once to see the selected region, as View #2.
   
   
6. Zoom in to the center once again, to create View #3, as seen in figure I.9 following:
   
   Figure I.9: A good view of the graph's interesting area
   

To edit a relation

1. Bring the relation window back up front, to be the active window: If some part of the window is visible, click on it once. Alternatively, go to GrafEq’s top Graph menu, drag the mouse down to Relation #1(Algebraic), and release the mouse to select the algebraic window for Relation #1.
   

   Hint: the relation window appears but not the prompt message, or the easy button floating window, the Active box in the relation window is selected, and the graph of Relation #1 is shown in the View #3 window.

2. Click once on the relation, to invoke its edit mode.
   

   Hint: the prompt message appears in the relation window, and the easy button floating window appears next to the relation window, the Active box in the relation window is cleared, and the graph of Relation #1 is not shown in the View #3 window.

3. Place the cursor just in front of “sin”, as seen in figure I.10 following:
   
   Figure I.10: Relation #1 being edited
   

   Tip: Move the cursor by clicking the mouse in the area, or by repeatedly pressing the left arrow key.

   Hint: The relation portion behind the cursor becomes unformatted. Don’t worry about it; it will be automatically reformatted as they pass to the left side of the cursor.

4. Erase the sqrt(x) term by either:
   • highlight the term with the mouse and cursor, then press the del key, or
   • press the delete key until the term is removed.
   
   
5. Enter x² as follows:
   1. Type in x from the keyboard;
   2. Go to the easy button floating window, and click the arrow adjacent to the algebra easy button set to open it;
   3. Click the a² button once, and the relation window will look like figure I.11 following:
      
      Figure I.11: Relation #1 edited
      
   
   
6. Move the cursor to the rightmost end of the relation, and see that the entire relation becomes formatted again, by:
   • clicking the mouse there, or
   • repeatedly pressing the right arrow key until the cursor is there, or
   • holding down the control key and pressing the right arrow key once.
   
   
7. Press the delete key three times to remove the 1/x term.
   
   
8. Enter x² one more time to complete relation editing, as seen in figure I.12 following:
   
   Figure I.12: Relation #1 edited
   

9. Press return to see the new graph plotted in the View #3 window.
   
   Figure I.13: Graph of the edited Relation #1
   

10. The new graph above does not look very interesting. But note that the scales show very small numbers, and remember that we had zoomed in twice to produce View #3; we will now zoom back out to take a look at the “bigger picture”.
    
    
11. Return to the zoom buddy, referring to the steps just described earlier when altering the viewport region, press the Zoom Out button to zoom out. The result of zooming out is intriguing. The following picture, View #7, is obtained after clicking the Zoom Out button four times.
    
    Figure I.14: Graph of the edited Relation #1 after four zoom-outs
    

To use constraints in defining a relation

• we can define the bounds in the Create View window, and that
• we can use the zoom feature of GrafEq.

1. Make the relation window the active window, referring to the steps just described earlier when editing a relation.
   
   
2. The prompt message at the bottom of the window states “....press tab to add a constraint.”. Press the tab key now.
   
   
3. Enter the constraint “0<x<3”, as seen in figure I.15 following:
   
   Figure I.15: Relation #1 with second constraint
   

4. Press return for the resulting graph, as seen in figure I.16 following:
   
   Figure I.16: Graph of Relation #1 with two constraints
   

1. press the mouse button on the File menu, then
2. drag the mouse down the pulldown menu to Close Graph, and release the button to close your graph.

To plot simultaneous systems

1. Under the File menu select New Graph, to get a new relation window.
   
   
2. Enter Relation #1 as seen in figure I.17 following, (with the help of the easy buttons for square operations):
   
   Figure I.17: Relation #1 for the simultaneous system
   

3. Press return after Relation #1 is entered correctly, to get to the Create View Window.
   
   
4. The default settings are fine, so press return or click on Create to create a graph.
   
   
5. Under the Graph menu, select New Relation, to get another relation window (entitled Relation #2).
   
   
6. Enter the second relation, referring to figure I.18 following, then press return.
   
   Figure I.18: Relation #2 for the simultaneous system
   

7. Open another relation window (entitled Relation #3), by selecting New Relation under the Graph menu.
   
   
8. According to figure I.19 following, enter the first constraint of the relation, press Tab, then enter the second constraint of the relation:
   
   Figure I.19: Relation #3 for the simultaneous system
   

9. Press return; your graph now shows all three relations in three different colours, layered with Relation #1 at the bottom, then Relation #2, and finally Relation #3 on top, as seen in figure I.20 following:
   
   Figure I.20: Graph of the simultaneous system
   

To blend graphs of simultaneous systems

1. Return to the view buddy, and return to colour mode by selecting Colour from the mode pop-up menu on the top of the floating window.
   
   
2. Click to select the Blend checkbox. The graphs for the simultaneous systems become blended, revealing individual graphs and intersections of the simultaneous system, as seen in figure I.21 following:
   
   Figure I.21: Blended graph of the simultaneous system
   

To determine a point’s coordinates

1. Return to the view buddy, and enter One Point mode by selecting One Point from the mode pop-up menu on the top of the floating window.
   
   
2. Place the mouse close to an intersection point on the graph; pin down the point with the help of the magnified view on the view buddy, as seen following:
   
   Figure I.22: Pin down an intersection point with the one point view buddy
   

3. Read the coordinates of the point from the view buddy.

To change the colours of the graphs

1. Access the colour mode view buddy again.
   
   
2. Click on the colour box of Relation #1. When the colour panel appears, drag the highlight box to choose another colour.
3. Repeat the process to choose colours for any other relations.

To determine distances between points

1. Access the Two Point mode of the view buddy. A line segment AB appears on the graph.
   
   
2. Use the mouse to drag points A and B to the two ends of the distance to be measured. See figure I.23 following as an example:
   
   Figure I.23: Measure distance between two points with the two point view buddy
   

3. Read the distances from the view buddy.

Figure I.24: Reward Relation

Many other features remain to be explored, such as:

• copying and pasting relations and graphs,
• making pages for printing,
• saving graphs to disk,
• using custom ticks,
• using one-key short-cuts, and
• customizing GrafEq for subsequent sessions.