Help - contents

•What is the program for?

•How to use Help

•Getting started

•Equations and transformations

•Marking sets

•Drawing graphs of functions

•Use of the mouse,   	   appearance of the mouse pointer

•Rough notes,   	   calculator,   	   description of the problem

•Working with a disk

•Printing,   	   print dialog box

•Options


•Getting started

•Menus

•On-screen buttons

•Windows

•Finishing work


•Equations and transformations

•Entering and editing expressions

•Solution of problems


•Marking sets - overview

•Sets on the number line

•Regions in the plane


•Saving problems to disk

•Loading saved problems

•The library

•Examples

•File selection dialog box


•Options

•Program configuration window

•Fonts


What does 2xy do?

Getting started

2xy is a computer program to help you with your math lessons.

It will enable you to:

•transform algebraic expressions,

•solve equations, inequalities and simultaneous equations,

•mark sets on the number line and the coordinate diagram,

•draw graphs of some functions.

You can work with 2xy completely on your own - you can use it with a clear conscience to help you with your homework.

On request, the program will always check the calculations performed so far, hence using 2xy you cannot get an incorrect solution! You can also ask the program for a hint or for mathematical help.

Version 2.0 for Windows™, IV 2000.


 How to use Help

While working with 2xy you can ask the computer either for Help with using the program or for Mathematical Help.

Help can be requested:

•in connection with the current operation, by pressing the appropriate keys F1 or ?

•on any subject, using the Help and Mathematical help menus

Words contained in the Help text which are underlined like this can be clicked on to open a smaller window with a more detailed description.

Click on words underlined like this to skip to a new Help chapter.

To return to where you were, use the Back button.

Use the Search button to find a Help chapter containing information on a desired subject.


Click anywhere to close the window


Click on the Back button or here


Getting started

What does 2xy do?   	   Finishing work

When the program starts, a window appears on the screen with a list of examples. The examples demonstrate the program's features and methods of solving problems. If you are using 2xy for the first time, you should definitely have a look at the first two examples!

Of course, you can close the examples window and start solving your own problem.

The program first waits to be told what calculations are to be carried out. You can do this by:

•using the commands on the File menu at the top of the screen, or

•clicking one of the buttons on the left-hand side of the screen.

You can enter a problem yourself in the on-screen window or load an example from disk.

During work there are also three helpful windows permanently open:

•Rough notes

•a simple calculator,

•description of the problem.


Menus

Getting started   	   On-screen buttons   	   Windows

To select a command from the menus at the top of the screen:

•click on the menu line,

•press the ALT key and then select the desired command using either

∗the arrow keys and ENTER, or

∗the keys corresponding to the letters underlined in the menu commands.

Contents of the menus appearing in the program:

File	Edit	Options	Mathematical help	Help
(description)	(description)	(description)	(description)	(description)


Select calculation type, change working window, save, load, exit


Commands used during calculations


Loading problems from disk, saving problems to disk, printing a solution


Change some program settings


Mathematical Help to assist with solving problems


Help with using the program


Equations and inequalities - to solve equations, inequalities and simultaneous equations

Transformations - to transform algebraic expressions

Marking sets on the number line and in the plane

Graphs of functions - to draw graphs

Close - exit the current calculation

Save - save the problem to disk

Save as... - save the problem using a new name

Load from library - load a problem from the library

Load user-defined file - load a problem saved by the user

Load example - load an example that demonstrates the program's features

Browse... - browse the entire solution in one window

Print... - print a solution

Report... - view a report on the problems solved so far

Exit... - exit the program

(window list) - list of windows open on screen


Fraction - insert a fraction line

Power - write the exponent in power notation

Root - write a root

Subscript - place a subscript after the name of a variable

Absolute value - the absolute value symbol

Per cent - the percentage sign

Delta - the Greek letter DELTA

Signs: - insert equality and inequality signs

Logical symbols: also set operations

Invert fraction - exchange the numerator and the denominator of a fraction

Block: - operations on selected parts of expressions

Undo - cancel the previous edit operation

Add a line below in a set of simultaneous equations

Add a line above in a set of simultaneous equations

Domain - insert a domain condition

Operation on both sides - perform an operation on both sides of an equation or inequality

Contradiction - the problem has no solution

Identity - the conditions given are satisfied everywhere

Ready to check - the solution is ready for checking

Final answer - the result obtained is the final solution

Substitute data - substitute data into the problem

Breakdown into cases: - separate the problem into different cases


Not equal

Less than or equal to

Greater than or equal to

Substitute a variable as an auxiliary variable


...is an element of...

Negation

Logical conjunction(AND)

Logical disjunction(OR)

Intersection of sets

Union of sets

Infinity

Set of real numbers

Empty set


Select all - select the entire contents of the window

Copy - copy the selected part of the expression to the clipboard

Cut - cut the selected part of the expression to the clipboard

Paste - insert the clipboard contents at the cursor position

Delete - delete the selected part of the expression


Add a new case - break down a problem into cases

New breakdown - change the way in which a problem has been broken down into cases

Exclude the current case - exclude the current case from the solution


Preferences... - change certain program settings

Fonts... - choose a font type to be used in the working windows

Printers... - select which printer to use

Library directory... - change the source directory for library problems

Example directory... - change the source directory for examples

Strict domain checking - causes the program to check the domain regardless of the solution

Hide keyboard - hides the on-screen keyboard

Show keyboard - displays the on-screen keyboard


Contents - displays a list of basic chapter topics

Topic search... - search for a desired Help topic

Hint - display a hint while solving problems


Contents - display a list of basic chapter topics

Topic search... - search for a desired Help topic

How to use help - the way of using Help and Mathematical Help

About... - display information about the current version of the program

More about 2xy - what can 2xy be used for?


On-screen buttons and function keys
On-screen buttons

Getting started   	   Menus   	   Windows

Button	Menu command	Button	Menu command	Key
	Strict domain checking		Save	CTRL+S
	Equations and inequalities		Load from library,  Load user-defined file	F3, F4
	Transformations		Load example	F5
	Marking sets		Help,  Mathematical Help	F1 ?
	Graphs of functions		Exit	ALT+F4
	Close


Solution of equations, inequalities and simultaneous equations


Transformation of algebraic expressions


Marking sets on the number line and in the plane


Drawing graphs of selected functions


Quxit the program


Close window or windows containing calculations


Save problem to disk


Load a problem from the library


Load a user-defined problem


Load an example that demonstrates the program's features


Help with using the program, in connection with the current window


Mathematical Help in connection with the current calculations


Windows

Getting started   	   Finishing work

When a particular type of problem has been chosen using a menu command or button, the following will appear on the display:

•A working window, in which calculations will be carried out or graphs drawn.

•A data window (if solution of equations or transformation of expressions has been chosen).

•An on-screen keyboard, with buttons which can be used when entering mathematical expressions.

Of course, we can make use of several different windows, for example:

∗open the equations and inequalities window,

∗solve an inequality in that window,

∗open the set marking window,

∗move the solution of the inequality to the new window,

∗and then mark the result on the number line.

It is also worth making use of the three permanently open windows: rough notes, calculator and description of the problem

To go to a particular window, use the mouse to click on it. You can also use the File menu, where the names of all currently open windows appear.

The working windows may be freely moved about the screen, resized or reduced to an icon, as is normally the case in Windows. It is also possible, using the mouse, to move the spliter [image] which divide the windows into two parts, and also change the position of the on-screen keyboard so that it does not cover up the calculations.


Finishing work

Getting started   	   Windows

Windows containing calculations can be closed using the button at the top of the frame (as is standard in Windows). You can also use the button [image] on the left-hand side of the screen or the command Close on the File menu.

Note:

If during solving equations or transforming expressions the problem was broken down into cases, then it is not possible to close just one of the windows. If you try to do it, all the windows will close.

If a change or addition has been made to the window being closed, then the program prompts for confirmation, suggesting that you save the problem to disk.

To exit the program, click the RED BUTTON on the left-hand side of the screen or use the Exit command on the File menu. Before exiting, the program displays information about problems solved and mistakes made.


Equations and transformations

In the transformations window you can operate on the algebraic expressions which you have entered.

In the equations window you can solve:

•equations,                 	   •inequalities,

•simultaneous equations,   	   •simultaneous inequalities.

The solutions of examples may contain:

∗numbers and unknowns,	   ∗fractions, including multi-layer ones,

∗brackets,                       	   ∗powers and roots.

In the equations or transformations window, first enter a problem, and then solve it yourself.

Performing calculations in 2xy is simple:

∗Type in the problem in the window and press ENTER,

∗Carry out a transformation and press ENTER,

∗Carry out a transformation and press ENTER,

∗...

∗Continue until you get your final result.

∗Then press ENTER without making changes in the expression

   or click the Answer button.

The program checks your working at each stage.


Entering and editing expressions

Window appearance and use    	   Solution of problems   	   On-screen keyboard

Mathematical expressions can be entered using the keyboard or mouse as well as the buttons on the on-screen keyboard. Menu commands can also be used.

When solving an example the following can be entered:

• basic operation symbols,

• numbers and letters (variables),

• brackets,

• fractions and quotients,

• powers,

• roots,

• sets and logical expressions.

It is also important to understand how to:

• move the cursor,

• delete numbers and expressions,

• select and move parts of expressions,

• move expressions using the mouse,

• insert data into a problem.


Solution of examples
Solution of problems

Window appearance and use   	   Entering and editing expressions   	   On-screen keyboard

After entering a problem, press ENTER  or the [image] button, or else double-click on the text.

Each time this key is pressed, the program:

•checks whether the calculations have been performed correctly;

•provides information about the type of mistake made, if any;

•if the transformations were done correctly, stores another stage of the calculations in memory, adding a tab at the top of the window.

With a little skill it is possible to solve the whole problem in one stage, although then it is easy to make a mistake. It is better, therefore, to make more frequent checks.

When the ENTER key is pressed, the problem is copied to the bottom of the window. The entire text in this lower part of the window is automatically selected, so it will disappear if you start to enter the next stage of the solution in its place. It is possible, however, to perform calculations by erasing redundant parts of the example and replacing them by the results of operations.

When solving problems it is also possible to:

• operate on both sides,

• substitute auxiliary variables,

• add conditions to the definition of the domain,

• break the problem down into cases,

• and ask the computer for a hint by pressing the SPACE BAR.

The final step in the calculation process is to give an answer. You can just press ENTER without making any changes in the expression. The program then displays the number of transformations performed and mistakes made.


On-screen keyboard

Window appearance and use   	   Entering and editing expressions   	   Solution of problems

This includes buttons which are necessary for entering and transforming expressions. By pointing at a button with the mouse pointer for a short time, you can see information about its function and corresponding keys.

The on-screen keyboard contains only the most indispensable buttons:

 keys used for entering expressions

 keys used for deleting and editing expressions

 keys used during the solution of problems

It is possible, however, to open extra panels:

 brackets,    [image] letters and digits     [image] logical symbols,

 cursor movement buttons,      [image] breakdown into cases.


Buttons used for entering expressions
Keys used for entering expressions

Entering and editing expressions   	   Solution of problems   	   On-screen keyboard

Basic keys and commands:

	+ , - , * , \	4 basic operations  the asterisk * denotes multiplication. \ key is used to write the division sign: /
	/	Fraction line
	^	Power
	@	Root
	_	Subscript following the name of a variable
	%	Percentage
	|	Absolute value
	< , = , >, ALT+< , # , ALT+>	Equality and inequality signs
	CTRL+=	Substitute auxiliary variable
	SHIFT+D	Add domain condition
	CTRL+\	Operation on both sides
	ALT+INSERT	Add a line above in a set of simultaneous equations
	INSERT	Add a line below in a set of simultaneous equations


To enter a fraction, start by pressing /  (fraction line). Next, using the UP/DOWN ARROW KEYS, move the cursor to the numerator and to the denominator of the fraction, entering the appropriate numbers or expressions there. The LEFT/RIGHT ARROW KEYS enable you to position the cursor in front of the fraction or after it.

The numerator and denominator may contain any expression, even fractions. When entering a complex fraction with several layers, always start with the main fraction line!

Note:

A number standing in front of a fraction is treated as its whole-number part, as long as the number, the numerator and the denominator contain only natural numbers. If a decimal point, minus sign, bracket, letter or any operation appears there, then the whole-number part is treated as a separate number, with a question mark ? between it and the fraction, indicating that an operation symbol must be inserted there.


When using power notation, to insert the exponent use the ^ key (  SHIFT+6  ), which raises the cursor by one line. After entering the exponent it can be brought back down using the arrow key.

A whole number, decimal, variable or any bracketed expression can be raised to a power.

Note:

The exponent may contain any expression using whole numbers,

addition, subtraction and multiplication (but not division) and brackets. To obtain fractional powers the root sign must be used.


The @ key enables you to insert roots. The root sign appears on the screen and numbers or expressions can be written under it. Using the arrow keys you can also move the cursor above the root sign and insert its degree, which can be any expression subject to the same limitations as the exponent of a power.


In 2xy all three kinds of brackets can be used:    (  ),      [  ],        {  }. Remember that the opening and closing brackets must be of the same type.

The brackets   (  )    and    [  ]    are also used to denote open and closed intervals.


In 2xy it is possible to enter numbers containing a maximum of 8 characters (including the decimal point). Any longer number will automatically be split into two using the question mark ?.  In this case the number must be shortened or an operation sign inserted.

If the exponent of a power or an operation sign is written in the middle of a number then that number will be split. The removal of a sign means that the numbers on either side are joined as one.

In the program only lower-case letters are used (except for the Greek letter Δ, obtained using CTRL+D).

A letter may be given a subscript, creating variable names such as [image] or [image]. To do this use the _  key, which lowers the position of the cursor. The subscript can be a single letter or digit.


Because in the absolute value notation the opening and closing

symbols are identical, in the program it is necessary to insert operation

 symbols between absolute values. For example:

|a|b|c|        could be read as

|a|*b*|c|      or      |a*|b|*c|

The solution of equations involving absolute values usually requires

 breaking down the problem into cases.


The following keys are used to insert logical symbols and set operations:

	SHIFT+E	...belongs to the set...		~ key	negation
	; key	semi-colon, used between the end-points of intervals		ALT+8 ALT+i	infinity (the program requires the insertion of + or -)
	SHIFT+Q	empty set		SHIFT+R	the set of real numbers
	& key	logical connective "AND".  You can just type "and"		SHIFT+V	logical connective "OR". You can just type "or"
	CTRL+^	set intersection		SHIFT+U	set union
Logical connectives (OR ∨ , AND ∧) can be displayed either using the symbols ∨ , ∧ or in words, depending on the setting Logical connectives (in symbols / in words).


Selecting and editing blocks

Entering and editing expressions   	   Solution of problems   	   On-screen keyboard

 To select a block, hold down the left mouse button and drag the mouse pointer over it, or use the arrow keys with the SHIFT key held down. If the cursor is then moved, this will undo the selection.

To delete a block press the DEL or BACKSPACE key. A block can also be dragged to a new location using the mouse.

It is also possible to select:

•A root sign itself:

press left arrow key with the SHIFT key held down (or BACKSPACE key)

when the cursor is at the beginning of the expression,

or press right arrow key with the SHIFT key held down (or DELETE key)

when the cursor is at the end of the expression,

Next you can press DELETE or BACKSPACE to delete the root sign, leaving the expression which was underneath.

Example:

	press DELETE		press DELETE again

•A fraction line together with the numerator or denominator:

press left arrow key with the SHIFT key held down (or BACKSPACE key)

when the cursor is at the beginning of the numerator or denominator,

or press right arrow key with the SHIFT key held down (or DELETE key)

when the cursor is at the end of the expression,

Then you can press DELETE or BACKSPACE to leave only the unselected part of the fraction.

Example:

	press BACKSPACE		press BACKSPACE again

Other buttons relating to operations on selected blocks are as follows:

	CTRL+A	select the entire contents of the window
	CTRL+C CTRL+INS	copy the selection to clipboard
	CTRL+X	cut the selection to clipboard
	CTRL+V SHIFT+INS	paste clipboard contents at the cursor position

You can easyli insert the selected block to the numerantor or denominator of a fraction, or under a root sign.

Just press / or @ key while the blosk is selected.

Examples:

	press @ key
	press / key;  notice that the cursor is after the block
	press / key;  notice that the cursor is before the block


The cursor can be positioned most quickly by using the mouse to click in the required position.

 When the mouse pointer is in a position where the cursor could be inserted it takes the form of a pencil.

The following keys are also used to move the cursor:

	TAB - to move between the top and bottom halves of the window PgUp, PgDn - to see other stages of the calculation
To see previous stages of the calculation, you can also simply click on the appropriate tab.

	arrow keys - these enable you to move the cursor within the current number (fraction, quotient etc.) and outside it.
	HOME and END - to move to the start or to the end of the current line.
	SHIFT+arrow keys - to select blocks of text

Tip:

It is a good idea to practice using these keys by loading a large example expression and moving around it.

We suggest you try the library problems included with the program.


Deleting and editing expressions

Entering and editing expressions   	   Solution of problems   	   On-screen keyboard

To delete single characters and selected blocks use the following keys:

	BACKSPACE deletes the character preceding the cursor
	DELETE deletes the character following the cursor
Note:

It is only possible to delete a fraction line when it is "short'' - ie. when the numerator and denominator contain only natural numbers.

A root sign can only be deleted immediately when there is no expression appearing under it.

In more complex quotients and roots, pressing the BACKSPACE or DELETE keys cuses the block to be selected

The expression is deleted when you press the key again, since the DELETE and BACKSPACE keys can also be used to delete selected blocks of text.

The following buttons are also very useful:

	CTRL+/	Inversion of fraction		CTRL+Z  ALT+BKSP	Undo


Use CTRL+Z or ALT+BACKSPACE to undo the previous operation on the current expression,

 eg. to restore a number after accidentally deleting it.


Use CTRL+/ key to exchange the numerator and denominator of a fraction.

 It is useful when transforming  multi-layer fractions.


Hints

Solution of problems

When performing calculations, press the SPACE BAR or click on the button [image] to see a hint relating to the operation on whose sign the cursor is positioned. You can also just click a right mouse button at the operation sign.

A hint can be obtained when the cursor is in the lower (working) section of the window. It should be positioned:

•on (more precisely before) one of the four operation symbols:  +,  -,  *,  /,

•on a number with a power,

•at the beginning of a fraction line,

•on a root sign,

•on a bracket (opening or closing),

•before an equality or inequality sign - for a hint on how to solve the equation.

Note:

Make sure that the expression is in mathematically correct form before pressing the space bar - the first thing the program does is check the notation!

The hint box looks like this:


Keys used while solving problems

Entering and editing expressions 	Solution of problems               On-screen keyboard

The following keys enable you to check whether your calculations are correct and to give an answer.

	ENTER	Ready to check
	SHIFT+ENTER	Final answer
	CTRL+ENTER	Exclude the case
	SHIFT+S	Substitute data
	SHIFT+i	Identity
	SHIFT+C	Contradiction
	SPACE BAR	Hint


The  ENTER  key should be pressed after entering the problem and after each stage of transformation - whenever you want the program to perform a check.

Each time this key is pressed, the program:

•checks whether calculations have been performed correctly,

•in case of error, informs the user what sort of error it was,

•if the transformation was done correctly, stores in memory another stage in the calculations, adding a tab at the top of the window.

If you press ENTER without making changes in the expression, the program will check the final answer.


Data for the problem

Entering and editing expressions   	   Solution of problems   	   On-screen keyboard

Numerical data relating to a problem is entered in a separate data window, which appears on the screen together with the calculations window. It is headed "Data and auxiliary variables", because definitions of auxiliary variables used in the problem are automatically transferred here. Initially the data window is minimized and can be restored if necessary.

Data can be entered in the same way as the problem itself. This must be done before the ENTER key is pressed for the first time.

Before giving a final answer to a problem involving data, the data values must first be inserted into the calculations window. To do this use the button [image] or press SHIFT+S. The data will be added to the domain definition.

If a parameter can take several different values, eg.

  x=1,        x=3,        x=16

then in the data window the logical connective OR should be used:

  x=1  OR  x=3  OR  x=16

After substituting data the solution must be broken down into cases and separate results obtained for each value of x.

When solving an equation or inequality containing parameters, eg.

    ax+b=0, where a, b are parameters,

it is helpful to enter the letters representing parameters in the data window, each on a separate line.

This lets the program know for which unknowns the equation is being solved for.

Note:

During 	∗transformations of algebraic expressions,

or 	∗if the program is set to strict domain checking,

the knowledge of data values can be taken into account only after substitution takes place. Until then all conditions should be retained, eg.

a≠0  when the data contains the condition a=2.

You can remove the redundant domain condition after substituting the data.


Sometimes, after breaking down a problem into cases,

 it turns out that one case is already covered by the others.

For example, the equation:

(x-2)(2x-4)=0

has two solutions:

x-2=0                    or                    2x-4=0

However, the second solution is the same as the first: x=2

In a case such as this, close the second case using CTRL+ENTER.

This will exclude the case from the solution.


Click the button [image] when the equation (inequality, system of simultaneous equations) being solved is always satisfied. The program will insert the word "identity" in place of the example and treat this as the final answer.


Click the button [image] when the equation (inequality, system of simultaneous equations) being solved is contradictory, having no solution. The program will insert the word "contradiction" in place of the example and treat this as the final answer.


The answer

Entering and editing expressions   	   Solution of problems   	   On-screen keyboard

In 2xy you decide for yourself when the solution is complete. If you have nothing more to do, press ENTER without making changes in the expression. You can also click the button [image] or press SHIFT+ENTER.

The program then checks the solution and decides whether it is in a suitable form for a final answer.

If something appears to be wrong then a dialog box appears similar to a hints window, requesting confirmation:

Click on YES in order to further modify the solution.

If you choose NO then the program must accept the answer in the form given.

The program will also accept the answers contradiction and identity.

Note:

If numerical data was entered for a problem, then the program requires that the data be substituted in before the answer is given.


Operating on both sides

Entering and editing expressions   	   Solution of problems   	   On-screen keyboard

When solving equations or inequalities we often apply mathematical operations to both sides of the equation (inequality). In 2xy these operations can be entered not only in the lower (working) section of the window, but also in the upper section, in which the previous stage of the calculations is stored.

To perform operations on both sides, click on the button [image] or press CTRL+\. A vertical bar will appear, after which should be written an operation symbol and a number or expression:

The equation or inequality in the lower part of the window must correspond to the equation (inequality) at the top after operating on both sides. Do not additionally move terms from one side to the other, swap the two sides over or change the order of the lines in a system of simultaneous equations.

Example:

The program checks whether these operations have been performed correctly the next time the ENTER key is pressed.


Auxiliary variables

Entering and editing expressions   	   Solution of problems   	   On-screen keyboard

Some more difficult problems can be solved with the help of auxiliary variables. These variables are defined using the ← symbol, which is obtained by clicking the button [image] or pressing CTRL+=, eg.

Note:

•Do not use the = sign in definitions of auxiliary variables

•The discriminant of a quadratic equation Δ is also introduced as an auxiliary variable.

A given auxiliary variable name can be introduced only once per problem. For this reason, after ENTER is pressed the definitions of new variables are moved from the working window to the data window, where we can see all the auxiliary variables introduced so far. In the working window the definition is copied as a new equation - with the = sign. You can remove it, or transform to a more convenient form.


The domain

Entering and editing expressions   	   Solution of problems   	   On-screen keyboard

When a fraction, root or complex expression is simplified during calculations, the domain of the problem often changes. To obtain an expression which is equivalent to the previous one you should add the appropriate conditions to the definition of the domain. This is especially important during transformations of algebraic expressions.

To add a domain condition click on the button [image] or press SHIFT+D. A Domain: line will appear above the expressions being transformed - here conditions defining the domain of the problem can be entered. Additional conditions specified in the original problem should also be given here.

Example:


Breaking down a problem into cases

Entering and editing expressions   	   Solution of problems   	   On-screen keyboard

Some problems have several different solutions, which it is convenient to treat separately.

Examples:

Keys used to break down a problem into cases are made available by clicking the button [image]. Above the on-screen keyboard there also appears a panel which shows the complete organization of the problem, eg.

Click on the icon for a particular case to go to the corresponding window.

Note:

If the problem contains more than one case, then it is not possible to close those windows individually. If you try to do it, all the windows will close.


The following buttons are used for breaking down a problem into cases:

	SHIFT+Y open a window for a new case
	SHIFT+X change the breakdown into cases (eg. combine two into one)

Note:

Windows can be closed and the breakdown into cases reversed before pressing ENTER to confirm the operation.


•The equation

 (2x-6)(5x+10)(x-4)=0

has 3 solutions:

2x-6=0 x=3	5x+10=0 x=-2	x-4=0 x=4

•To solve the equation:

|x+1|-2x=5

we must consider two cases:

x+1≥0                                            x+1-2x=5 -x=4 x=-4	x+1<0                                            -x-1-2x=5 -3x=6 x=-2


Drawing graphs of functions

Graphs window

In 2xy you can draw the graphs of the following types of function:

•linear

•quadratic

•linear rational functions (hyperbolas)

Note:

The function

 in accordance with the above formula is written as

                      .

To draw the graph of a function, you must:

∗enter the formula for the function,

∗give the coordinates of several points on the graph

∗mark the points on the coordinate diagram.

In one window can be drawn the graphs of several functions of the same variables. To do this, press ENTER after finishing the first graph. This creates a new tab at the top of the window and lets you enter the formula of another function.


The formula of a function is entered in the upper part of the window. The required expressions are built up in the same way as during transformations or solution of equations.

 Remember:

•on the left-hand side of the equality sign there should be a single variable (representing the value of the function),

•the right-hand side may contain only one variable (the argument of the function),

•the formula must be written in one of the three forms allowed by the program.

When the formula has been entered, press the ENTER key. Before this select a color for the graph using the mouse and the colored semi-circle at the bottom of the window.


Entering coordinates of points

Graphs window   	   Graphs of functions - overview   	   Entering formulas for functions

The coordinates of points on a graph are entered in a table which appears after the formula for the function has been given and ENTER pressed. The upper row of the table contains values of the argument, while the lower row contains the corresponding values of the function. Remember that all points must be different.

The coordinate values can be whole numbers, fractions, decimals or even irrational numbers. They are written in the same way as in the equations and transformations window.

The table contains the minimum number of columns needed to fix the position of the graph. It is possible, however, to add extra columns using the button [image] (or INSERT key - to add a column to the right of the current one) or [image] (ALT+INSERT - to add a column to the left of the current one).

According to what sort of function is being considered, the table contains:

•for linear functions - space for the coordinates of 2 different points,

•for quadratic functions - space for the coordinates of the vertex of the parabola

  and 2 other points,

•for hyperbolas - a column containing

∗the argument value for which the function is undefined (the denominator is 0),

∗the value which the function cannot take

Example:

  and one column for the coordinates of a point on the graph.

After all required coordinates have been entered, press ENTER. You can then start to mark points on the coordinate diagram.


The graph of the function defined by the following equation:

  has its vertex at the point:


If the function has the equation

then the argument cannot take the value 0

    x≠0

The value of the function can never be equal to 0

    y≠0


The Graphs window and its use:


To mark a point with given coordinates on the diagram:

•click on the appropriate column of the table,

•move the mouse pointer to the desired point on the coordinate diagram

 (if necessary you can change the position of the origin or the scale of the diagram

 using the buttons in the bottom left-hand corner of the window),

•click the left mouse button.

When all the points have been marked the computer will automatically connect them

and produce the graph.


Marking sets (regions)

Marking sets window   	   Sets on the number line   	   Regions in the plane

2xy lets you mark sets on the number line and in the plane. This enables you to solve graphically inequalities and simultaneous inequalities in 1 or 2 variables.

In the regions window, start by entering the conditions (equations and inequalities) you wish to illustrate. The conditions should be entered on one line, joined using the logical connectives ∧ (AND),  ∨ (OR) and brackets.

The conditions given can contain:

With 1 variable	With 2 variables
• equations and inequalities of the type x=a  , x≠a  ,  x<a  , where a is a number;	• linear equations and inequalities, of the type x<a  ,          where a is a number; y<a  ,          where a is a number; y>ax+b ,  where a,b are numbers;
• double inequalities of the form a<x<b  ,               a,b  numbers;	• double inequalities of the form a<x<b  ,                 a,b numbers; ax+b<y<cx+d  ,         a,b,c,d numbers;
• numerical intervals, eg. x∈(a;b] x∈(-∞;a) x∈[a;+∞) and combinations of them, eg. x∈(a;b]∩(c;d) x∈(a;b)∪[c;d]	• inequalities written using intervals, eg. x∈(a;b] y∈(-∞;a) y∈[ax+b;cx+d) or combinations of these, eg. x∈(a;b]∪(c;d) y∈(ax+b;e)∩[cx+d;+∞)
Note that:

∗round brackets ( ) are used to denote open intervals, whereas square brackets [ ] are used for closed intervals,

∗a semi-colon is used to separate the end-points

∗the infinity sign ∞  must be preceded by + or -.

• statements of the form:

x∈ℜ                    (x is any real number)

x∈∅      (the set of values of x is empty)

What happens next depends on the number of variables which appear in the given conditions.


Marking sets on the number line

Marking sets window   	   Marking sets - overview   	   Regions in the plane

After entering the set of conditions and pressing ENTER you should:

• click on a particular equation, inequality or interval.

The selected condition is shown inside a box whose color will have been chosen earlier using the colored semi-circle.

Illustration

If there is only one condition, then it is selected automatically after you press ENTER.

• mark the corresponding set on the number line, by clicking on one of the end-points, moving the mouse to the other end-point and then releasing the mouse button.

∗In the case of a single inequality, the interval's other end-point will be -∞ or +∞

The mouse button should be released at the left-hand or right-hand edge of the window, in the region marked with the infinity sign.

Illustration

∗If the condition corresponds to an empty set, eg. x∈(1;-2)  then click on the [image] button at the bottom of the window.

∗If the condition contains = or ≠ then it is sufficient to click on just one number.

The selected interval is shown in a box.

Illustration

• use the mouse to specify which of the end-points belong to the interval (in the case of a closed interval or weak inequality) and press ENTER.

Illustration

Show all the equations, inequalities and intervals on the number line in the same way.

Illustration

Next, using the mouse, indicate on the bar just above the number line which points and intervals satisfy the given conditions. Then press ENTER.

Illustration

Often it can be seen from the diagram that the set of conditions could be written more simply. In this case press ENTER again and enter the new version of the conditions at the top of the window. When you press ENTER again the program will check whether the two versions are equivalent.


The conditions are entered in the upper section of the window. Expressions can be built up in the same way as during transformations and solution of equations.

The following panel of buttons on the on-screen keyboard is useful here [image].

Remember to make sure that the conditions entered have the correct form.

After entering the conditions, press ENTER.


Marking sets in the plane
Marking regions in the plane

Marking sets window   	   Marking sets - overview   	   Sets on the number line

After entering the set of conditions and pressing ENTER you should:

• use the mouse to select a particular equation, inequality or interval end-point.

The condition will be shown in a box, whose color will have been chosen earlier using the colored semi-circle.

Illustration

If there is only one condition, it is selected automatically when ENTER is first pressed.

• in the table at the bottom of the window enter the coordinates of points on the line which forms the boundary of the region satisfying the selected condition.

Illustration

If the line has already been drawn (as the boundary of another region) then the program automatically moves on to the next point.

• mark the points on the coordinate diagram.

Illustration

• click on the region lying on the appropriate side of the line, or on the line itself in the case of a weak inequality. Press ENTER when ready.

Illustration

Show all the equations, inequalities and interval end-points on the plane diagram in the same way.

Illustration

Next click on all the points, line segments and regions which satisfy the conditions given. Press ENTER when ready.

Note:

If a line passes through a selected region, remember also to select the segment of the line which lies in that region!

Illustration

It can often be seen from the graph that the set of conditions could be written more simply. In this case press ENTER again and enter the new version of the conditions at the top of the window. When you press ENTER again, the program will check whether the two versions are equivalent.


The Marking sets window and its use:


Use of the mouse

In 2xy, the mouse can be used for:

•moving and re-sizing windows,

•pressing buttons on screen.

You can also:

•position the cursor by clicking on a selected number or expression,

•select part of an expression by dragging the mouse over it with the left button held down,

•holding down the left mouse button, move the selected block to a new position or to another window,

•holding down the right mouse button, move a single number (or an entire quotient or root)  to a new position or to another window,

•use the right mouse button to get a hint,

•mark the position of points on the number line and in the coordinate diagram,

•double-click in the working window instead of pressing ENTER.

The appearance of the mouse pointer indicates what can be done with the mouse at a given moment.


To move a selected expression using the mouse:

•Move the mouse pointer onto the expression - the pointer should change shape

•Press the left mouse button to "pick up" the expression. A box automatically appears around the object being dragged.

To move a single number which has not been selected, press the right mouse button.

•Keeping the mouse button held down, move the pointer to the desired position. Dragging a number over a window automatically causes it to become active.

•Release the mouse button. The expression will be copied to the current cursor position in the chosen window. While doing this, observe the appearance of the mouse pointer!

•If the mouse button is released when not inside any window, then the only effect is to remove the box from around the number.

An expression selected as a block can also be copied to the clipboard (CTRL+C or CTRL+INS) and pasted anywhere (CTRL+V or SHIFT+INS).

Note:

An expression being dragged using the mouse is stored by the program in the form of a sequence of keystrokes needed to create the expression. For example, if a mixed number is moved to the exponent of a power, then only the integral part will be inserted there, since the exponent cannot contain fractions.


Appearance of the mouse pointer

The appearance of the mouse pointer depends on where it is and provides information about what you can do with it.

Other than the standard Windows arrows, which show that windows can be moved, the following pointers appear in the program:

	Basic pointer for the whole screen. Clicking the mouse button will have an effect only when the pointer is on a button.
	This pointer appears when the mouse is positioned on a button. Click to press the button.
	The pencil indicates a position where the cursor could be placed. To position the cursor click the left mouse button.
	The question mark appears when you press the right mouse button on the expression. It indicates the possibility of getting a hint. Just release the button to get a hint.
	This pointer shows that the number or expression can only be dragged using the mouse and copied to a new position.
	This appears when a number is being dragged, when the cursor cannot be placed at the current mouse position. This means that releasing the mouse button will cause the expression to be placed where the cursor was previously.
	When a number is being dragged, this indicates a position where the cursor can be placed. Releasing the mouse button will cause the expression to be dropped here.
	When an expression is being dragged, this indicates a position to which the expression cannot be copied. Releasing the mouse button here means that the dragging operation is canceled.
	This "sights" pointer is used for marking points on the number line and coordinate diagram.
	This indicates that the position of the bar separating parts of the calculations window can be adjusted.
	This pointer appears on the on-screen keyboard to show that it can be moved. It can also be seen on the colored semi-circle which is used to select graph colors.


Rough Notes is a simple graphics editor. It includes buttons for drawing and entering formulas:

Examples of use

Note:

The Rough Notes text is not checked by the program.


Saving a problem to disk

Loading a previously saved problem

To save a problem to disk, use the Save command from the File menu, the function key CTRL+S or the diskette button on the left-hand side of the screen. The problem may be saved at any time.

When a problem is saved for the first time, a name for it must be given. A file selection dialog box will therefore appear on the screen. If the program already knows the name of the problem then it will be saved immediately after CTRL+S is pressed.

To save the problem under a new name use Save as... on the File menu.


Loading a previously saved problem

Saving a problem to disk

To load a problem, use the Load command on the File menu. You can load:

•a problem you entered yourself, by using the user-defined command, pressing F4 or clicking on the school bag button;

•a library problem, by selecting from library, pressing F3 or clicking on the bookshelf button.

Next select the name of the problem in the file selection dialog box.

After loading a problem you will start from exactly the same stage in its solution as you were at when you saved it.

When loading a new problem you can close all open windows or leave them open, adding new calculations windows.

Note:

The program loads only the window contents, not their dimensions. Often it is necessary to start by suitably resizing the windows.


File selection window
File selection dialog box

Saving a problem to disk   	   Loading a previously saved problem

The file selection dialog box contains:

•A panel containing a one-line textual description of the problem and the algebraic expression that describes it (by default this is the problem as entered in the first open working window).

You can edit the descriptions while saving the problem.

Click the Preview check-box to disable/enable the preview of the problem description.

•A list of disk directories. Double-click on a directory to open it.

•A list of disks installed on the computer.

•A list of files in the currently open directory. Double click on the name of a problem to select it for saving or loading.

•A list of permitted file extensions: *.LIB , *.2xy , *.*

•A line on which the name of a problem can be entered.

•Three buttons:

OK - saves or loads the file (you can also press ENTER instead),

CANCEL - closes the window (you can also press ESC)

HELP - displays information about the window.


The library

Loading a previously saved problem	Examples

The program includes a library of over 100 problems of various types and levels of difficulty. These can be useful, for example, when revising for tests.

Library problems are saved as files with the extension .LIB. This is a reserved extension - you cannot use it for saving your own problems (unless the LIBRARY CHANGE ALLOWED setting has been changed). If you wish to save a problem which you loaded from the library, the program requires that you give it a new name.

Note:

Library examples contain only the text of the problem, ready for checking. Before starting to solve the problem you must press ENTER !

The library window contains a list of problems of the selected type. Each problem is described by a mathematical expression an a line of text.

The items of the combo-box labelled Problem class correspond to sub-directories of the library folder.

You should select a problem by clicking on ot and press the Open button.

Categories of library problems:

• Equations in 1 unknown -  EQUATION sub-directory.

• Inequalities in 1 unknown - INEQUALI sub-directory.

• Marking sets on the number line and in the plane - REGIONS sub-directory.

• Simultaneous equations in 2 unknowns - S_EQUAT sub-directory.

• Simple transformations of algebraic expressions - TR_EASY sub-directory.

• Transformations of rational expressions - TR_RATIO sub-directory.

• Transformations of expressions with roots - TR_ROOTS sub-directory.


Examples

Loading a previously saved problem	The library

As well as the library of problems for you to solve by yourself, the program also contains a set of fully worked examples. They are stored in the EXAMPLES folder as *.XMP files.

The example select window appears when the program starts - unless you click the check-box in the window, labelled "Do not show this box at start". To load an example at a later stage, click the [image] button.

After loading the example the program automatically starts to enter and solve the problem, displaying descriptions of each action. You cannot direct the solving process, but you can

∗ make the presentation go faster by pressing ENTER,

∗ stop and resume the presentation with the SPACE bar,

∗ interrupt the presentation (with the ESC key) and continue the solution yourself.

The example window window contains a list of problems of the selected type. Each problem is described by a mathematical expression an a line of text.

The items of the combo-box labelled Problem class correspond to sub-directories of the library folder.

You should select a problem by clicking on it and press the Open button.

Categories of example problems:

• how to edit expressions - the EDITING sub-directory,

• how to solve problems - the SOLVING sub-directory,

• how to mark regions and draw graphs - the GRAPHS sub-directory.

It is especially recommended that you load the first two examples of the Solving type when you work with 2xy for the first time.


Printing

While using the program you can print out the problem being worked on. This makes it easier to copy the solution into your exercise book. You can also ask your teacher if he or she is willing to accept the program's printouts as completed homework.

The printout includes a description of the problem, calculations and graphs from all open windows (together with all transformations) and a report on the number of mistakes made. It is also possible to print out your rough notes.

If an example is too wide for the paper, then it will be carried over to the next line, marked by a dotted line.

When you want to print, use the Print command on the File menu. The print dialog box will appear on the display.


Print dialog box

Printing - description

The print dialog box contains:

•a list of available printers, from which you can choose the type you wish to use,

•a box in which you can enter a title to be printed above the problem,

•an option allowing you to print your rough notes,

•an option for black-and-white or color printing. To change the setting, click on the button or its label.

•an option for vertical or horizontal paper orientation. To change the setting, click on the button or corresponding picture.

•the following  buttons:

OK - to begin printing. You can also press ENTER instead.

CANCEL - to close the window without printing. You can also use the ESC key.

HELP - displays this text,

SETUP... opens a system dialog box for setting printer configurations.


Options

We can change the program's settings using the following commands from the Options menu:

•Preferences... -opens a program configuration dialog box, in which the following settings can be changed:

∗Logical connectives (in symbols / in words)

∗Keyboard(large/small)

∗Time limit

∗Library change allowed

∗Mouse at button

•Strict domain checking - if this is ticked it means that the program is set for strict domain checking.

•Library directory,

 Example directory - set the source directory for library problems and examples.

•Hide keyboard - hides the on-screen keyboard.

•Show keyboard - displays the on-screen keyboard in the bottom right-hand corner.

•Fonts... - opens a fonts dialog box for changing the fonts used in the working windows.

•Printers... - opens a system printer settings dialog box.

When you exit the program, its current settings are saved in a file called 2xy.INI in the Windows folder. This file will be reloaded next time the program is run. It is also possible, therefore, to configure 2xy by editing the INI file.


Program configuration window
Program configurations dialog box

Options - overview

This window lets you change the following settings:

•Logical connectives (in symbols/in words) - determines how the logical connectives AND, OR are to be shown

•Keyboard (large/small)

•Time limit - a limit on the number of transformations performed by the program

•Library change allowed - determines whether library problems can be modified.

•Mouse at button - in dialog boxes.

To confirm your choices click on OK or press ENTER. Use the  CANCEL button or ESC key to close the window without making any changes.


Time limit - this option means that the program can only perform the specified number of transformations (measured in appropriate units). If you wish to carry out very complex calculations and you have a fast computer and plenty of time, then you can increase the limit.


Logical connectives (in symbols/in words) - enables you

to choose how the logical connectives ∧(AND), ∨(OR) are to be displayed.

Note:

This setting does not affect the keys used to enter the connectives, only the way

they appear on the screen.


Keyboard (large/small) - this changes the size of the on-screen keyboard.

 	The larger keyboard is easier to use on a high-resolution screen.


Mouse at button - when a dialog box is opened, the program positions the mouse

 pointer on the default button. This is a convenient, though non-standard, option.


Library change allowed - This option allows you to save files with the extension .LIB,

 and hence to modify the problem library. Recommended mainly for teachers.


Fonts

Options - overview

Choose this menu command in order to change the fonts used in the working windows. A dialog box will appear in which the following settings can be made:

•font name - select from the list or enter using the keyboard;

•typeface: bold or italic - click on appropriate label;

•character size - use the buttons on one of the two fields marked "Size"  (the program uses two character sizes: normal and reduced, used for example in power notation).

•line thickness - used in fractions and for root signs.

The program uses two types of font:

•letters - used for variable names and other labels.

•mathematical symbols - these should be consistent with the system font SYMBOL.

To confirm the choice of fonts for letters and symbols, click on OK. Click on CANCEL to close the window without making any changes.


Strict domain checking

2xy always checks whether the solution and domain  of the problem (equation, inequality or expression being transformed) are correct. However, this can be done in the following ways:

•During transformations of algebraic expressions the program rigorously checks the domain of the expression regardless of the data values.

example

•When we are solving equations and inequalities, the program substitutes numerical data into the problem immediately and takes the domain conditions into account only in as far as they affect the final result.

However, we can enable strict domain checking by pressing the [image] button at the top left corner of the screen.

Then 2xy

∗does not use the information about numerical data before they are formally substituted;

∗checks the identity of the domains before and after the transformation. If the domains are not identical, the Warning dialog box appears. You can ignore the warning or return to the working window and improve the domain definition.

example 1

example 2


We solve the equation:

 (strict domain checking is on)

The second term on the left-hand side can be canceled, provided that x≠0.

The equation takes the form:

  x+1=2

When ENTER is pressed, the program warns us about the change of the domain.

We verify that x=0 is not a solution of the equation which results from canceling

the fraction. Cancellation does not therefore affect the solutions of the equation and

can be carried out without adding a domain condition. We can ignore the warning.

The result is correct. The final result is:

  x=1


We solve the equation:

 (strict domain checking is on)

The second term on the left-hand side can be canceled.

The equation takes the form:

  x+1=1

When ENTER is pressed, the program warns us about the

change of the domain.

This warning cannot be ignored, since  x=0 is a solution

of the equation which results from canceling the fraction.

We should add the domain condition:  x≠0.

Next we reduce the numbers. The result: x=0 does not

satisfy the domain condition. The equation has no solution.


We find the value of the expression:

for a=3

In this case the problem could be restated as follows: "Find the domain of the

expression, simplify it and calculate its value for a=3".

We can cancel the fraction dividing numerator and denominator by a and b.

However we must add the domain conditions a≠0 and b≠0.

	After canceling, we substitute the value of a and obtain the answer 3. The condition a≠0 is redundant if we know that a=3, but it can only be deleted after the substitution has been made.


Click this menu item to view the whole solution in a single window - like printed on paper.

The [image] button in the answer dialog box can also be used to open the browse window.