@EIf you want to change the first graph in the file, which is quite often the only one, you can press 'Enter' to modify it quickly without having to choose the type. The dialogue with which you will be presented will be almost the same as the one used to create it, so see the previous section for more information.
Alternatively, if you need to change the type or edit a graph other than the first, you can either choose 'Graphs' on the Toolbar, or 'Graphs' from the 'Graph' menu.
If there is at least one graph function in the file, you will be presented with a dialogue giving a list of those functions and five buttons.
Choosing 'Close' will close the dialogue, and, when necessary, redraw the screen to take into account any changes made.
Choosing 'New' will create a new graph (See previous section).
Choosing 'Delete' will delete the selected graph after prompting you. (You will only be prompted if the 'Confirm delete' option is selected in the Global Preferences dialogue).
Choosing 'Modify' will allow you to modify the graph currently selected. This is done in almost the exact same way as creating a new graph, so see the previous section for more information.
The remaining choice, 'Copy' acts the same as 'New', except that the type choice is bypassed, and the details from the selected graph are copied to a new function.
BAs well as solving functions graphically (by either tracing them until they intercept the x-axis or by using the 'Find point' function), Grapher can solve simulataneous and quadratic equations numerically, finding both real and complex roots
Simultaneous
Select 'Simultaneous' from the 'Maths' menu. Type in the relevant values or expressions and choose 'Solve'. Grapher will then display the values of x and y which fit the equations if possible.
Quadratics
Select 'Quadratics' from the 'Maths' menu. Type in the values of a, b and c (they can be expressions) into the fields then choose 'Solve'. Grapher will then display the roots for that equation (either repeated, real, or complex).
AGrapher interacts like any other Series 5 application. New files can be created from the System screen, and Grapher can be run from the Extras bar.
If you are unsure of how to use the Series 5, read your user manual or the system-wide help.
Grapher is SHAREWARE. Registering will greatly increase the speed of graph drawing and will remove the nag screens. See the section "Registering" below for more details.
OThere are four ways to create a new graph:
1. Choose 'New' from the toolbar.
2. Choose 'Add new graph' from the 'Graph' menu.
3. Chose 'New' from the 'Graphs' dialogue (see next section).
4. Press one of the 'x', 'y', 'o', 't', or 'i' keys
The first three choices will bring up the 'Choose graph type' dialogue, where you can choose the type that you wish the new graph will be. The available types are:
Cartesian y=f(x) (key 'x')
Cartesian x=g(y) (key 'y')
Polar (key 'o')
Parametric (key 't')
Inductive (key 'i')
Once you have selected the type you want, choose 'OK', or 'Cancel' if you decide you do not want a new graph.
Choosing 'OK', will presented you with a second dialogue where you set the details of the graph which will depend on the type chosen, but for all types, there will be the colour, width, and style (explained below), and for each, you choose 'Add' to draw the new graph, or 'Cancel' to return back to the previous dialogue.
However, if you pressed one of the keys shown above, you will have bypassed the type selection dialogue and have gone straight to the dialogue for the particular key pressed.
Graph styles
There are five graph styles to choose from: line, dashed line, cross (X), cross (+), and dot. These are pretty self-explanatory, but when using 'dot', it is a good idea to increase the width, otherwise it is very difficult to see.
Cartesian y=f(x)
These are Cartesian graphs where the y coordinate depends on a function of the x coordinate.
Type the function into the 'f(x) =' field.
You can also choose an equality for the graph. 'y =' will draw a single line following the graph function, while 'y <' or 'y >' will shade the area which does NOT fulfill the equation, so that with multiple inequalities, a white area is left which fulfils all the equations.
Cartesian x=g(y)
This is as for the above, but here the x coordinate depends on a function of the y coordinate.
Polar
These are graphs of the form 'r=h(o)', i.e. a polar graph where the radius depends on a function of the angle (o was chosen as the variable name, because it is quite close to the Greek letter theta which cannot easily be represented on the Series 5).
Type the function into the 'h(o) =' field.
You can also choose an equality for the graph - but remember, Grapher shades the area that you don't want.
Parametric
These are graphs of the form 'x=f(t), y=g(t)', i.e. the x and y coordinates are determined by independent functions of an independent parameter, t.
Type the function f into the 'f(t) =' field, and the function g into the 'g(t) =' field
Parametric graphs cannot have inequalities.
The ranges for polar and parametric graphs can be set by either:
a) choosing 'Set range' in the dialogues when you create the graphs, or
b) choosing 'Set polar/parametric range' from the 'Graphs' menu.
Both these dialogues operate in much the same way as the 'Set range' dialogue described in the section 'Moving and Zooming'. Note that expressions are allowed in this dialogue, for example, 2PI.
Inductive graphs
These are graphs where the next value depends on a function of the current value, which are plotted so that the current value is plotted up the y-axis, and the number of iterations (times that the function has been calculated) along the x-axis. It is necessary to specify the first value of t, t0, because inductive functions require a starting value.
Type the function of t and n% into the 'f(t,n%)' field, and the value of t0 into the 't(0)' field. You also need to specify the range of n%, the current iteration number. This always starts at 0 and goes to the value in the 'Range of n%: 0 to ' field. It is possible to use 'n%' in the function, so for example, 't=n%', with t(0)=0 would be a straight line through the origin. However, since this function does not depend on itself, it cannot really be classed as inductive.
Inductive graphs can have inequalities.
NOTE: You can have a maximum of eight graphs of any combination types in any one file.
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)NMoving Around
You can move around the graph using the cursor keys. One press will take you one quarter of a screen in the direction of the arrow pressed.
You can centre on the origin by choosing 'Centre on origin' from the 'View' menu.
Zooming
Zooming in
There are three ways to zoom in:
1. Choose 'Zoom in' from the 'View' menu.
2. Press the + key on the left side of the screen.
3. Press the pen onto the screen.
You will be presented initially with some crosshairs in the centre of the screen (or wherever you pressed the pen). These can be moved with the cursor keys, or by pressing the pen down. Use the Ctrl key to make the crosshairs move faster.
Having positioned the crosshairs, you have two choices:
Firstly, you can zoom in on that point. This is achieved by pressing Enter. The graph will zoom to the quarter of the screen surrounding the crosshairs.
The alternative is to select a box to zoom the screen to. To do this, you can either use the cursor keys with Shift held down (and optionally Ctrl to move faster), or you can drag the pen from the start to the finish. When you do this, the crosshairs will disappear and be replaced by a box. When using the pen, if you hold down Ctrl, the box will take the same relative dimensions as the screen. This means that you can preserve a 'square' range when zooming in, provided that you start with one.
When you have positioned the box to your satisfaction, press Enter and the screen will zoom to fill that box.
Zooming out
To zoom out, either press the - key on the left side of the screen or choose 'Zoom out' from the 'View' menu.
The screen will zoom out so that what previously filled the entire screen now fills the middle quarter.
Previous Zoom
If you make a mistake when zooming, or simply want to return to the previous zoom setting, choose 'Previous Zoom' from the 'View' menu.
Setting the Range
If you want to set a specific range either choose 'Set' from the 'View > Range' menu or select 'Range' from the Toolbar.
You will be presented with the 'Set range' dialogue. This has a pair of three fields, one for the x values, and one for the y values. The minimum values express the values at the bottom and left edges of the screen, and the maximum values those at the top and right edges of the screen. The steps determine the distance between the scale markings on the axes (but will not affect how graphs are drawn).
In each of these fields, you can not only just enter a number, but also expressions. This can be especially useful when dealing with trigonometrical functions, since multiples of PI can be used.
The minimum range is -1000, the maximum 1000, and the range of the step is >0 to 500.
If you choose 'Initial', the settings will revert to their initial values, i.e -10 to 10, step 1 for both x and y.
If you choose 'Square', the x ranges will be adjusted so that one unit on the x axis will represent the same amount on the y axis.
Choosing 'Set' will set the range and redraw the screen at the correct scale, while choosing 'Cancel' will leaving the range settings as they were.
Alternatively, you can quickly set the range to specific values. These features are accessed from the 'View > Range' menu. Choosing 'Initial' will have the same effect as choosing 'Initial' from the 'Set range' dialog, but will be applied immediately. Choosing 'Square' will make the current display square (see above). Finally, 'Trigonometrical' will set the x-range to the equivalent of -360 to 360 (depending on the angle format setting), and the y-range to
1.2 to 1.2. This is useful when drawing graphs of trigonometrical functions.
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EGrapher can perform numerical approximations derivative of a function and to the definite integral of a Cartesian graph function.
For information about finding the derivative, see the 'Tracing a Graph' section above.
First, create a cartesian graph function in the usual way. Then choose 'Integration' from the 'Maths' menu. A dialogue will open where you set the details of the integration.
Choose the cartesian graph function that you want to integrate.
Then choose the method of integration. There are two choices - 'Simpson's Rule' and 'Trapezium rule'. Simpson's rule is generally more accurate, but there may be some circumstances where you need the trapezium rule.
The 'Range from' and 'Range to' fields are where you specify the range for the integration. These can be expressions as well as just numbers.
Finally, the 'Number of Strips' determines the accuracy of the approximation. The larger the number, the longer it will take, but the more accurate it will be, and vice versa. Since Simpson's rule requires an even number of strips, this number is actually doubled so that you need not worry about this. If you put the number of strips as zero, Grapher will automatically work out an appropriate value for this based on the current zoom level.
When you have entered the details choose 'Integrate' to perform the approximation, or 'Cancel' to cancel the operation.
When it is finished, a new dialogue will appear reporting the result.
PBGrapher has six constants which can be used in any graph function.
To use one of them, enter into an expression 'Kn' where 'n' is a number from 1 to 6.
For example 'K1x^2' would be x squared times the first constant term, and 'K6^K4' would be the sixth constant raised to the power of the fourth constant.
To alter the constants, choose 'Constants' from the 'Maths' menu. This will bring up a dialogue with six fields into which you can enter expressions for the new values of the constants. Choose 'Set' to set the new values or 'Cancel' to close the dialogue leaving the values intact.
HChoose 'Trace graph' from the 'Graph' menu to activate trace mode. To leave this mode you press Escape.
Here you will be presented with some crosshairs on the first graph, which may be off-screen depending on the function. To change the graph being traced, use the up and down cursor keys. To move along the chosen graph, either use the left and right cursor keys, or drag the pen on the screen. You can use the Shift key to move more slowly, and the Ctrl key to move faster. At the bottom of the screen will be displayed the current x and y coordinates.
Note that the crosshairs may go off screen depending on where the graph function leads them.
When tracing inductive graphs, it is only possible to step forwards, i.e. use the right cursor key.
Please note that when you trace a graph containing a
, Grapher will only allow you to trace the positive part.
Autotrace
It is also possible to have Grapher automatically trace the graph. To do this, press the Space bar while in Trace mode. The crosshairs will automatically follow the graph. You can press the left and right arrows to alter the speed of the autotrace. When you are finished, press Escape to return to manual trace mode.
Differentiation
It is possible to get an approximation to the derivative at a point on all but Inductive graph functions. To do this, move to the correct position as usual in Trace mode, then press 'd'. You will be presented with a dialogue giving the approximate derivative and second derivative at that point. If you wish, you can choose for a tangent graph to be created at that point, provided you have less than eight graphs in memory at that time (i.e. enough space for another graph function).
For parametric graphs, you will get the overall derivative at the point, in other words, dy/dx or dx/dy, and not dx/dt and dy/dt. Flipping between dy/dx and dx/dy ensures that the gradient can always be represented; otherwise vertical and horizontal lines could be a problem. Parametric graphs will not give the second derivative.
You cannot add a tangent to a polar graph.
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GThere are two types of preferences in Grapher, the File and the Global preferences.
File preferences are saved along with the current file and affect the display. Global preferences are saved in a single, separate file, and these affect such things as default graph colours, sound, etc.
Titles
It is also possible to give a Grapher file a title, and this also comes under the 'Preferences' submenu. Choosing 'Title' will allow you to set the text and the position on the screen. If you set the text to nothing, then no title will be shown.
File Preferences
To open the dialogue, choose 'Preferences
File' from the 'Tools' menu.
Here you can set:
whether the axes are drawn,
whether a grid is drawn,
the colour of the axes,
whether the axes are labelled,
whether graphs are labelled,
and the resolution for Cartesian graphs.
All but the resolution are pretty much self-explanatory.
The resolution basically allows you to trade off speed for accuracy, '1' being the most accurate but slowest, and '16' being very quick, but not very accurate. This only affects Cartesian graphs, since polar and parametric graphs have their own dialog for this ('Set polar/parametric range'), and inductive graphs always go in steps of 1.
Global Preferences
These are available from 'Preferences
Global' from the 'Tools' menu.
Here you can set:
the default properties of graphs,
whether sound is on,
whether help messages are shown (and the status messages of 'Trace', 'Zoom', 'Find'),
whether Grapher confirms the deletion of graph functions,
whether the grid will be on when you create a new file.
When sound is on, a click will be sounded as soon as all the graphs in memory have been drawn. This can be useful to notify you when you have several complex graphs being drawn.
The help messages are little reminders of what to do that appear in several places, such as when you start a zoom: 'Select area and press Enter to zoom'.
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ZBIntroduction
Grapher is an all-purpose function graphing application. Each Grapher file can contain eight overlayed graphs, each with its own colour, width and style. The graphs can be Cartesian, polar, parametric, or inductive, with or without inequalities. You can trace graph or find a point on the screen, for example to locate the intersection of two graphs. You can integrate and differentiate functions, and solve quadratic and simultaneous equations. Grapher also has advanced zooming capabilities, is fully configurable, and can be operated from the keyboard without the pen if you wish.
Welcome to Grapher V1.36
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Using Grapher
Creating a Graph!
Modifying Graphs
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Moving and Zooming
Tracing a Graph
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AFWhat is shareware?
This application is not free software - it is shareware. This means that you can share the unregistered version amonst your friends (provided the original zip file is kept intact), but if you find yourself using it after the trial period of two weeks, you must register it. Registering will give you five things:
1) faster graph drawing
2) the removal of the nag screens
3) free upgrades to later versions
4) free support via email
5) peace of mind and a sound conscience in the knowledge that you are supporting shareware authors and the Psion community.
The software as it stands is functionally complete, but the registered version runs faster than the unregistered version, and the nag screens will be removed.
How do I register?
There are two ways to register: by post or via RegNet on the Internet. Post is cheaper, but will take longer and is difficult outside the UK.
To register on RegNet, go to
http://www.swregnet.com/1952p.htm
find Grapher from the list of programs, and choose to register it. This will cost 30 US dollars, and you will need a credit card.
To register by post, send me 15 UK pounds, or 25 US dollars (see the Contacting Me section for my address), preferably as a cheque, with your details including name, postal address, and e-mail address.
When you have registered, I will send you a code which can be used to unlock the software. This can be done by choosing 'Register' from the 'Tools' menu, typing in your name and code, and choosing the 'Register' button. There is no need to download or install the software again, just type in this code.
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GGChanges between version 1.35 and 1.36:
Version 1.36 incorporates a few bug-fixes, and a few new features:
1. You can evaluate in Trace mode
2. The mode you are in (i.e. trace, zoom, or find) will be displayed at the top of the toolbar instead of the filename to remind you that you are in these modes.
Changes between version 1.31 and 1.35:
Version 1.35 is mainly a maintenance release, fixing several bugs, notably
should now work properly, and I have fixed a few things in this help file. I have reduced the unregistered restrictions, and have generally tidied Grapher up. Constants are now accessed through the much shorter 'Kn', rather than the long and e cumbersome cnst(n), though this still works for compatibility. Grapher file titles should now not get overdraw.
New features include LOGB(base,x), support for
and
operator, and quick setting of initial, square, and trigonometrical ranges.
I have also made Grapher work much better when being run first time - gives a welcome message.
Changes between version 1.25 and 1.31:
Several bugs have been fixed, especially cartesian derivatives will now work properly.
The main new features are
1. More functions supported - cosh, sinh, tanh, cot, sec, cosec, and more - see Expression Formats and Functions.
2. The hotkeys for 'Find point' and 'Number Formats' have been swapped.
3. Quick modification of the first function - just press Enter.
4. Different line styles - dashed, dotted, and crossed.
5. Inequalities for polar and inductive graphs.
6. Derivatives for polar and parametric graphs.
7. Graph labels are now in bold.
8. A comma will now work properly as the decimal point if the Control Panel in the system screen has been set up correctly.
9. Better Trace display.
10. Simultaneous equations
11. Smoother drawing of graphs.
For more detailed descriptions, see the sections above.
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-AI would like to thank all those who have helped me develop Grapher, especially Steve Litchfield, and Matt Millar, and David Glover who helped me with debugging and provided some new ideas.
I would also like to thank all those who have registered Grapher or reported bugs - your support is invaluable.
GA quick guide to the syntax of Grapher can be obtained by choosing 'Syntax' from the 'Tools' menu.
The expressions which can be used in Grapher follow most of the basic algebraic rules, e.g. 2x becomes 2
x, (x^3)(x^2) becomes (x
), and 3cos(4x
3) becomes 3
cos((4
Note that the standard precedence rules apply, i.e. since 3
4x becomes 3
x, this works out as (3
4, and not 3
For square and cubic powers, you can use Ctrl
178 and Ctrl
179 to get the '
' and '
' characters. For other powers, you can use ^, e.g. x^1.5.
Functions
All functions in Grapher must have brackets around their operators in order to work, for example, LOG(10).
Grapher supports all the trigonometrical functions and there inverses:
cos(x),sin(x),tan(x)
sec(x),cosec(x) or csc(x),cot(x)
cosh(x),sinh(x),tanh(x)
To get the inverses of these, prefix them with an a. For example:
acosh(x)
You can use the
character (Ctrl-177) to indicate to Grapher that you want both the positive and the negative of the following to be drawn. For example:
SQR(25-x^2)
would draw the two halves of the circle, radius 5, with centre at the origin.
There are two constants built in, in addition to the user defined ones. These are E and Pi. To use E, type E^(x) where x is the power to which you want E to be raised. To use Pi, simple type PI.
As well as LN(x), natural logarithm of x, and LOG(x), log to base 10 of x, you can use LOGB(base,x) which will take the logarithm to the supplied base of x.
In addition to these, all the mathematical functions of OPL (the inbuilt programming language of the Series 5) can be used, for example SQR(x), square root, and INT(x), the integer part of x.
The units for trigonometric functions can be set by choosing 'Formats' from the 'Tools' menu. Currently radians and degrees are supported, though gradians are not. You can also choose the accuracy used throughout Grapher, for example in the axis labels and the find point information.
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AHGraphs
Pressing 'Esc' while a graph is being drawn will cause the plotting to stop. If you want the screen redrawn, you can choose 'Redraw display' from the 'View' menu to update the screen to the correct state.
If an error occurs while a graph is being drawn, then a dialogue will open telling you what was wrong. You will then be given a chance to correct it. If however, you want to delete that graph, choose 'Cancel' and it will be removed.
Screenshots
Although not directly supported by Grapher, it is possible to take screenshots of Grapher files so that they can be incorporated in other documents and possibly printed. (This does not only work in Grapher but in any application). To do this, get the display as you would like it, turn off the toolbar (or leave on if you wish), and press Ctrl-Shift-Fn-S. This wil bring up a system dialogue to allow you to save the screen. Once you have saved the screen, enter your document and create a Sketch object. In Sketch choose 'More
Merge
in' from the 'File' menu, select 'EPOC picture' as the filetype, and enter the name of the screenshot file that you have just saved. Then return to your document by choosing 'Done', and you should find the Grapher screenshot incorporated. It is probably a good idea to delete the temporary screenshot file.
Files
Grapher supports multiple files, in the same way as any other Series 5 application. If you do not know how this works, please read your user manual or the system-wide Help.
Toolbar
If you want to have a larger area for viewing the graphs, you can toggle the toolbar on or off by selecting 'Show toolbar' from the 'View' menu.
In Trace, Zoom, and Find modes, the respective name will be placed in the title of the toolbar to indicate the current mode.
About Grapher
You can find out the current version number and other information about Grapher by choosing 'About' from the 'Tools' menu.
Exiting Grapher
If you wish to exit Grapher, choose 'Exit' from the 'File' menu. All the details will be saved to the current file, so when you next load this file, it will appear just as you left it.
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Equation Solving
5 Constants
Preferences
Miscellaneous
Expression Formats and Functions
REGISTERING0
Acknowledgements
Contacting Me
My postal address is:
Jamie Shotton,
34 Kirk Close,
Oxford,
ENGLAND,
OX2 8JN
My email address is:
jamies@iname.com
My homepage is at:
http://www.geocities.com/SiliconValley/Bay/6740/
You can register online at:
http://www.swregnet.com/1952p.htm
New features in Version 1.36
@There are two types of preferences in Grapher, the File and the Global preferences.
File preferences are saved
along with the current file and affect the display. Global preferences are saved in a single, separate file, and these affect such things as default graph colours, sound, etc.
Titles
It is also possible to give a Grapher file a title, and this also comes under the 'Preferences' submenu. Choosing 'Title' will allow you to set the text and the position on the screen. If you set the text to nothing, then no title will be shown.
File Preferences
To open the dialogue, choose 'Preferences
File' from the 'Tools' menu.
Here you can set:
whether the axes are drawn,
whether a grid is drawn,
the colour of the axes,
whether the axes are labelled,
whether graphs are labelled,
and the resolution for Cartesian graphs.
All but the resolution are pretty much self-explanatory.
The resolution basically allows you to trade off speed for accuracy, '1' being the most accurate but slowest, and '16' being very quick, but not very accurate. This only affects Cartesian graphs, since polar and parametric graphs have their own dialog for this ('Set polar/parametric range'), and inductive graphs always go in steps of 1.
Global Preferences
These are available from 'Preferences
Global' from the 'Tools' menu.
Here you can set:
the default properties of graphs,
whether sound is on,
whether help messages are shown,
whether Grapher confirms the deletion of graph functions,
whether the grid will be on when you create a new file.
When sound is on, a click will be sounded as soon as all the graphs in memory have been drawn. This can be useful to notify you when you have several complex graphs being drawn.
The help messages are little reminders of what to do that appear in several places, such as when you start a zoom: 'Select area and press Enter to zoom'.
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Equation Solving
5 Constants
Preferences
Miscellaneous
Expression Formats and Functions
REGISTERING0
Acknowledgements
Contacting Me
My postal address is:
Jamie Shotton,
34 Kirk Close,
Oxford,
ENGLAND,
OX2 8JN
My email address is:
jamies@iname.com
My homepage is at:
http://www.geocities.com/SiliconValley/Bay/6740/
You can register online at:
http://www.swregnet.com/1952p.htm
New features in Version 1.36
JJChoose 'Trace graph' from the 'Graph' menu to activate trace mode. To leave this mode you press Escape.
Here you will be presented with some crosshairs on the first graph, which may be off-screen depending on the function. To change the graph being traced, use the up and down cursor keys (the function you have selected will be displayed in the top right corner). To move along the chosen graph, either use the left and right cursor keys, or drag the pen on the screen. You can use the Shift key to move more slowly, and the Ctrl key to move faster. At the bottom of the screen will be displayed the current x and y coordinates.
Note that the crosshairs may go off screen depending on where the graph function leads them.
When tracing inductive graphs, it is only possible to step forwards, i.e. use the right cursor key.
Please note that when you trace a graph containing a
, Grapher will only allow you to trace the positive part.
Evaluate
If you want to know the value of a function at a known x, y, o, or t value (argument), then simply press Enter in Trace mode. The function will be displayed along with the current tracing position. You can then type in your argument and press Enter. The function will be evaluated and you will be reported the result. You can then either close the dialogue, or set the trace position to the argument you just supplied (if out-of-range then it will be brought back).
Autotrace
It is also possible to have Grapher automatically trace the graph. To do this, press the Space bar while in Trace mode. The crosshairs will automatically follow the graph. You can press the left and right arrows to alter the speed of the autotrace. When you are finished, press Escape to return to manual trace mode.
Differentiation
Grapher can calculate an approximation to the derivative at a point on all but Inductive graph functions. To do this, move to the correct position as usual in Trace mode, then press 'd'. You will be presented with a dialogue giving the approximate derivative and second derivative at that point. If you wish, you can choose for a tangent graph to be created at that point, provided you have less than eight graphs in memory at that time (i.e. enough space for another graph function).
For parametric graphs, you will get the overall derivative at the point, in other words, dy/dx or dx/dy, and not dx/dt and dy/dt. Flipping between dy/dx and dx/dy ensures that the gradient can always be represented; otherwise vertical and horizontal lines could be a problem. Parametric graphs will not give the second derivative.
You cannot add a tangent to a polar graph.
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Welcome to Grapher V1.36
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Using Grapher
Creating a Graph!
Modifying Graphs
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Moving and Zooming
Tracing a Graph
Calculus$
@AI would like to thank all those who have helped me develop Grapher, especially Raphael Nottebohm, Steve Litchfield, and Matt Millar, and David Glover who helped me with debugging and provided some new ideas.
I would also like to thank all those who have registered Grapher or reported bugs - your support is invaluable.
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