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OPERATORS AND FUNCTIONS
Basic operators: Other operators:
──────────────────────────┬───────────────────────────────────────────────────
+ addition │ [( )] Parentheses: may be nested to any extent;
- subtraction │ parser WON'T distinguish between ( and [
* multiplication │ ; Separate halves of a parametric equation
/ division │ ' Make rest of the equation a comment
^ exponentiation │ {m, n} Specify domain, where 'm' is the start of
= normal equation │ the domain and 'n' is the end. Either end
──────────────────────────┤ may be left open by omitting 'm' or 'n'.
Functions supported: │ <, > less-than or greater-than inequality
──────────────────────────┼───────────────────────────────────────────────────
abs absolute value │ exp Euler's number to the given power (e^x)
acos arc cosine │ int greatest integer ([x] is not supported)
asin arc sine │ ln natural logarithm
atan arc tangent │ log logarithm base 10
asec arc secant │ sec secant (1/cos)
acsc arc cosecant │ sin sine
acot arc cotangent │ sinh hyperbolic sine ┌─────────┐
cos cosine │ sqrt square root │PgDn for │
cosh hyperbolic cosine │ (or sqr) │more help│
cot cotangent (1/tan) │ tan tangent └─────────┘
csc cosecant (1/sin) │ tanh hyperbolic tangent
CONSTANTS AND VARIABLES
Variables Usage
───────────────── ──────────────────────────────────────────────
x, y rectangular coordinates
r, t r and theta in polar coordinates
x, y, t x and y as functions of t in parametric form
t, x, dx, d2x... solves up to fourth-order ODE
(or x, y, dy...) dx is actually dx/dt in dx/dt = f(x,t)
a user-settable free variable; can specify range
and step rate to calculate several values
Constant Value
───────────────── ──────────────────────────────────────────────
b, c user-settable free variables (constant)
d converts degrees to radians = π/180
e Euler's number = 2.718...
pi (or p) π = 3.14159...
NOTE: by default, all trig functions work in radians, not degrees. You
can convert using the constant d: e.g. sin(45d) = sin (pi/4)
cos (x*d) = cosine of x, in degrees (you will need to change the range
of x to 0 to 360 to get the full graph) Press PgDn for more help.
GRAPHING CARTESIAN FUNCTIONS
Here are the ground rules for normal Cartesian equations:
+ Always use exactly one "=" per equation
+ Always use exactly one "y", which need not be isolated
+ Always put some sort of expression on both sides of the "="
+ An expression consists of any mathematically meaningful combination of
decimal numbers, binary operations like + and *, parentheses, functions,
and variables or constants.
+ Spaces are completely optional, except where they serve to separate
alphabetic identifiers.
+ The order of operations is the standard algebraic left to right of:
Functions, Parentheses, Exponents,
Multiplication and division, Addition and subtraction
Graphmatica supports implied multiplication of variables and constants, but
variables and other alphabetic identifiers such as functions and constants
MUST be separated by a space or '*' or they will not be identified correctly.
Examples: 3x 5(2x+3) x pi x*cos(x)
Call a function by tying the function's name followed by an argument. If the
argument is more complex than "x", enclose it in parentheses to be safe.
Examples: cos x log (2x) sin (3x^2-5)
Press PgDn for more help.
SPECIFYING THE DOMAIN
Graphmatica lets you specify the domain of each equation independently, so you
do not have to use the range function to change the default domain each time.
To specify a domain, add anywhere in your equation the expression:
{ m, n }
where 'm' is the start of the domain and 'n' is the end. If you want the
domain to start at the default start, leave 'm' out. To leave the end of the
domain open, leave out 'n'. So if the range on-screen is (-10,10), specifying
a domain of "{ ,5}" will graph from -10 to 5, and one of "{-4, }" from -4 to
10. To graph a parametric equation, you MUST specify a domain that is closed.
The domain is parsed like any other expression, so you can use any operator or
function as well as numbers and the constants d, e, and pi. However, you can
not use variables in a domain. Examples: {0, 2p} { -3, log 2}
USING FREE VARIABLES
The free variables 'b' and 'c', which you can define yourself, make it easier
to "play around" with the exact shape of a curve without editing the equation.
If you want, you can type in a value manually using a format similar to
the domain specifier described above. (See manual for details)
Examples: {b: 1} {c: -1/2} {b: -pi/4}
If you don't include this value, Graphmatica will prompt you for it.
Press PgDn for help on function families.
GRAPHING FAMILIES OF FUNCTIONS
The free variable 'a' lets you specify not just a single value, but a range of
possible values that it can take. This lets you easily graph families of
functions or level curves of a 3-D surface. For instance, "y = a*cos(x)" will
graph cosine curves of varying amplitudes, and "x^3+y^2 = a" will draw level
curves of the surface "f(x,y) = x^3+y^2".
When you use 'a' in an equation, Graphmatica will prompt for the needed values
and insert them for you: the start of the range, the end of the range, and the
amount to step by. Graphmatica starts by graphing the function with 'a' set to
the start of its range, and then increments 'a' by the step value and draws
another graph until 'a' exceeds the end of its range. (You can give a negative
step value as long as the end of the range is less than the start.) The
program does not put any limit on the number of curves in the "family" you can
graph, but this feature uses memory rapidly, so try to limit 'a' to about 10
iterations.
MODIFIYING FREE VARIABLE VALUES
The Set Vars command in the Point menu allows you to change the value of 'a',
'b', or 'c' instantly in all equations without actually editing any of them.
After you select a variable and provide a new value for it, Graphmatica will
find all the equations using the variable, make a copy of each using the new
value, and then graph them automatically.
Press PgDn for more help.
DRAWING THE GRAPH
Once you have entered an equation that parses successfully and pressed enter,
Graphmatica will proceed to graph it. To interrupt a graph when the computer
is in the process of drawing it, hit any key and the program will display on
the bottom line the message:
"PAUSE at x=#, y=#. Press ESC to quit, any other key to restart.. "
where # indicates the x and y coordinates you stopped it at. If you mistyped
the equation and want to fix it, just press ESC. Be patient! Graphmatica may
need a while to produce a quality graph on a slower machine. To speed up the
graphing, you may want to select a lower Fineness value.
When the graph for your equation is complete, the blinking cursor will
reappear in the equation editor. If you'd like to type in a completely
different equation, press ESC to clear the input field. If you'd rather modify
the last equation, go right ahead; it's already stored in the redraw queue.
Or modify any previously entered equation by pressing the down arrow key (or
clicking on the down arrow next to the editing field) to drop down the
equation queue listbox and then selecting an equation from the redraw queue.
You can then redraw this equation by pressing enter, or modify it to create a
new equation. To access Graphmatica's other features, press Alt to raise the
menu, or click on the menu with the mouse.
Press PgDn for help on inequalities.
GRAPHING INEQUALITIES
You can graph rectangular-coordinate inequalities by replacing the '=' with
'<' or '>' for many simple functions and relations. This feature is presently
only available for cartesian graphs. (Also, since the free variable 'a'
implies that more than one curve will be drawn, equations containing 'a' are
incompatible with inequalities, and the parser will not accept any inequality
that uses 'a'.)
The region that solves the inequality is hatched in with the graph color. (The
curve itself is not actually dotted to indicate a strict inequality, however,
so '<' is effectively "less than or equal to".) In most cases, asymptotes are
detected and a boundary added there as appropriate, so graphs like "y < tan x"
and "xy > 1" are drawn correctly. In addition, the valid domain of the
function being graphed is detected automatically, so "y > log x", for
instance, shades only the first and fourth quadrants.
To accommodate intersecting regions, the graphing routine alternates between
'\' and '/' hatching. Best results will be obtained when you graph no more
than two inequalities on the same screen; a region MAY SHADE INCORRECTLY if
another graph of the same color intersects it. If you have problems with
shading "leaking out," increasing the fineness should help.
Press PgDn for help on polar graphs.
EXTENDED GRAPH FUNCTIONS: POLAR GRAPHS
Polar coordinates are a fundamentally different approach to representing
curves in 2-dimensional space. If you have never used polar coordinates and
want to understand them, you should read Section III in the manual first.
To make a graph using polar coordinates, we calculate a distance to plot
out from the origin as we let theta ("t") sweep around in the positive
direction. The default domain is 0 to 2pi (the first complete circle in the
positive direction), but you can easily change these values using the
T range function [see View help file]. Polar graphs are entered in the
equation editor just like normal graphs. The only difference in what you type,
and how Graphmatica detects a polar graph, is that you must use the variables
't' and 'r' instead of 'x' and 'y'. The restrictions are still the same: you
can have one and only one instance of the dependent variable 'r' but it can be
located almost anywhere in the equation. Watch as your graph is drawn; often
the direction it is going is as important as the figure it draws. (In a
"double" equation of "r^2", first the positive and then the negative roots are
drawn; they should be drawn simultaneously but it isn't practically possible.)
The x and y coordinate ranges and the range for theta are completely
independent; in normal Cartesian graphing, Θ's value is irrelevant, and in
polar graphing, Θ controls the domain of the graph, but the x and y ranges
still control the physical screen you see. If you want to change your view of
a polar graph, use the scale or range functions just as you would normally.
Press PgDn for help on parametric graphs.
EXTENDED GRAPH FUNCTIONS: PARAMETRIC GRAPHS
(For a full explanation of parametric graphing, read Section III in
the manual. This section just shows how to enter parametric graphs.) In
parametric graphing, the cartesian x and y coordinates are calculated based on
a third variable (the "parameter" of x and y) called 't' (not to be confused
with the 't' used to represent theta). T is allowed to increase from the start
of the domain you specify to the end. At each value, the functions x(t) and
y(t) are calculated to give an (x,y) coordinate which is drawn. Graphmatica
then connects these points to form a smooth curve -- if something you graph
looks jagged, you probably need to adjust the fineness. (See Options help)
To enter a parametric graph, you need to remember four basic parts: the
x(t) and y(t) functions, the semicolon between them (this is how Graphmatica
knows you're entering a parametric graph), and the domain for t.
semicolon y-function
x-function -> x = 2t ; y = 2t^2 {-10, 10} <- domain
You don't need to solve for x and y (5x=t is OK), but only one x and one y can
appear in the whole equation, and "double" equations like "x^2=t" are NOT
supported (if you enter them only the positive root will be found). You MUST
specify a domain for each parametric equation! Some curves (like those based
on sin and cos) work best over a {0,2pi} domain, like polar graphs. Others
match the default domain of normal graphs better. If you estimate the domain
wrong, abort the graph and edit it. Press PgDn for help on differentials.
GRAPHING APPROXIMATIONS OF ORDINARY DIFFERENTIAL EQUATIONS
To solve a first order ODE, include the differential "dx" (which represents
dx/dt) in your equation. If you do NOT include a "domain" ("{ , }"), the
program will draw a slope field for dx/dt = f(x,t), with t as the horizontal
axis and x the vertical. If you do include a "domain" {m, n}, the equation
will be graphed as a specific solution to the initial-value problem x(m)=n by
Runge-Kutta approximation. Graphmatica also solves 2nd-4th order IV problems;
to specify higher order derivatives, use the variables d2x, d3x, or d4x.
Remember that for an nth order equation, you need to give n+1 initial values.
You can type these into the equation using the notation described above; the
order of values is t, x, dx, d2x, d3x. Thus d2x + x = 0 {0,0,1} graphs a sine
curve as the solution to "d^2x/dt^2 + x = 0" for x = 0 and dx/dt = 1 at t=0.
You can use the notation dy = f(y,x) if you prefer; both sets of variables
are automatically recognized as differential equations.
You can also specify an IV by moving the crosshairs with the arrow keys or
mouse. Choose Set IV from the Point menu, move the cursor to the desired
location, and press enter or click to add the IV point to the equation. For
higher-order ODEs, you can also specify the first derivative using the cursor.
After you select the IV point, Graphmatica draws a rubber band from it to the
current position until you click on a second point. The coordinates of the
first point are used as the t and x values, and dx/dt is set to the slope of
the line between the two points. If required, further initial values will be
requested via dialog boxes. Press PgDn for help on Error Messages.
ERROR MESSAGES
Eighteen error messages may be encountered when graphing. Thirteen of them
are fatal; the equation cannot be graphed and you must edit it. They will make
the computer beep so you know there is a problem. The other five apply only to
specific point(s) for which a y-value cannot be generated. They will not
appear unless turn the Warnings option on, and then appear silently.
[Note that all warning messages which refer to the variables 'x' or 'y'
will actually be 't' or 'r' when you are dealing with a polar equation.]
Please type an equation (or select from the listbox); then press ─┘ to graph.
You pressed enter on the graph line without selecting or typing in
an equation.
No equals sign or more than one found. Press any key to edit the equation.
To be a valid and graphable, your equation must include exactly one equals
sign ['=']. If you get this error, you either left out the '=' or
accidentally typed two or more of them. For parametric equations, there
must be an '=' on each side of the dividing ';'.
One or both sides of equation evaluated to nothing. Please edit your equation.
Make sure there is some sort of expression on each side of the
equation. Obviously, an entry like "y=" can't produce a meaningful graph.
Press PgDn for more error messages.
ERROR MESSAGES continued
Inequalities only supported for rectangular equations. Replace < or > with =.
Inequalities cannot presently be evaluated for polar, parametric,
or differential equations. You may still be able to draw the graph if you
can express the inequality in rectangular form.
Found bad operation or mismatched parentheses. Press any key to edit...
You equation had different numbers of open and close parens, was missing
an operand for a binary operation, or included something else the parser
couldn't digest. Examine your equation carefully and fix whatever
seems to be the problem.
Initial value must be on screen to graph an ODE. Modify grid or equation's IV.
Since an initial-value approximation that doesn't start in the visible
grid may never encounter it, Graphmatica tries to avoid this situation.
Simply move the IV point or modify the grid so that it is on screen.
Not enough initial values to draw graph. Make sure there are # and try again.
Although 1st-order ODEs may be graphed as slope-fields without IVs, higher
order ODEs require initial values for t, x, dx..., up to the derivative
one order less than the highest one in the equation. Make sure to specify
the right number of valid IVs.
Press PgDn for more error messages.
ERROR MESSAGES continued
Missing argument to a binary operation or function. Please edit your equation.
The parser couldn't find any identifiers or expressions to use as one of
the operands to a binary operation (like '+') or as the argument to a
function, like cos(x). All functions require one argument following the
function name.
Found unknown variable or function. Press any key to edit the equation.
Check that your equation contains only valid identifiers (see the
functions and variables tables in the Graph help file) and that you
separated each of them with an operator, space, or some other punctuation.
No dependent variable or more than one 'y' found. Please edit your equation.
Although Graphmatica can isolate ONE 'y' variable and graph some
relations, it cannot graph an equation without a 'y', like "x=4". It also
cannot perform the factoring needed to isolate the variable 'y' (or 'r')
when it occurs more than once (i.e. "x=y^2+3y"). If you can adjust the
equation so it uses only one 'y', do so; otherwise it can't be graphed. In
parametric graphing, this message may also indicate that no 'x' variable
was found in the x(t) equation.
Press PgDn for more error messages.
ERROR MESSAGES continued
Can't find the inverse of this function of y. Press a key to edit equation.
You tried to graph an equation like "int(y)=x" or "abs(y)=x" for which y
cannot be isolated by taking the inverse of the function. The functions
which cannot be isolated are "abs", "cosh", "sinh", "tanh", and "int". If
you can't adjust the equation so this error does not occur, it is not
graphable.
Parametric equation requires that you specify domain! See 'Graph' help file.
You typed in a parametric equation (or accidentally hit the semicolon) and
neglected to include a closed domain [like {1,6}]. Because the diversity
of parametric equations makes it hard to pick a default domain, you have
to include one with each parametric graph. See Sections I and III in the
manual for help on entering a domain and parametrics.
Domain did not evaluate to a constant value. Press a key to edit equation.
The domain you entered either could not be parsed, or was found to contain
a non-constant identifier, like x or y. Valid domains must have at least
one side of the range defined and can't contain variables, although any
other expression that evaluates to a constant is OK.
Press PgDn for warning error messages.
WARNING ERROR MESSAGES
Overflow at x=#.##.
Some function or operation generated a number too large to fit into an
eight-byte floating point variable. The point at x=#.## was not graphed.
Division by zero at x=#.##.
At #.## your equation attempted division by zero so the point was skipped.
Can't raise a negative number to a fractional power. [x=#.##]
Due to the possibility of getting an even root of a negative number, the C
Library pow() function refuses to process any arguments like these. The
portion of your graph (if any) where the base is not negative or the power
is not fractional should be graphed perfectly. This error also occurs when
you take the square root of a negative number with the "sqr" function.
Can't find the logarithm of a negative number. [x=#.##]
The natural logarithm (ln) and base 10 logarithm (log) functions are
defined only on x greater than zero.
Domain error: asin/acos functions defined only on -1≤x≤1. [x=#.##]
The arcsine (asin) and arc cosine (acos) functions are only defined
between -1 and 1 (the range of the sin and cos functions).
──────────────End of Graphing help. Press ESC to return to menu.──────────────