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# How To Draw Function Graphs (Mathematics / Analysis)

Empirical plotter for mathematical functions and calculus.

To view these instructions on the plotter page, click next to the instructions link. Click to hide them. Click on the slider   | | | | | | | | |   to change the frame size.

Content:
Syntax
Constants
Functions
Basic functions
Trigonometric and hyperbolic functions
Non-differentiable functions
Probability functions and statistics
Special functions
Programmable functions
Iterations & fractals
Differential and integral equations
Calculate single value
Gimmicks & Fun

With openPlaG mathematical function graphs can be drawn. Those images may be used at will. Up to three graphs can be shown in one image. For this, input fields for three formulas are available. For the proper use of the plotter, it is advisable to activate JavaScript.

Draw plots the image of the graph, Reset sets all values back and Standard sets back display settings.

## Syntax

The formulas f(x), g(x) and h(x) can contain the following symbols:

x Function variable x
0-9 Numbers, e.g. 123.45 Input values are hardly restricted (as long as you don't use numbers with more than 300 or so digits). The output value (result) at a non-log scale can only be up to 100000 (and -100000), this is also the maximum value for both axes.
Very large numbers can be written like 2.5E20 for 2.5*1020, very small ones like 3E-10 for 3*10-10. Decimal digits are exact up to a number of 12.
. , Point or comma as decimal separator, e.g. 1.5 or 1,5
( ) [ ] { } < > Brackets, e.g. {[(1+x)/(2-x)+1]*3}/(2*x^2) , are allowed in any amount. Each opened bracket must be closed again. What kind of bracket you use doesn't matter.

# as separator for formulas with multiple input values, e.g. scir(x#2)
asy Vertical asymptote for a given fixed value, e.g. asy(1) or asy(e)
Q Substitution for a self-defined formula.

#### - Basic arithmetic operations

+ Plus, e.g. x+1
- Minus, e.g. 1-x
* Times, this can only be omitted, when standing between a number and a letter. E.g. you can write 2x instead of 2*x, but not xsin(x) or ex.
/ : Divided by, e.g. 1/x or 1:x

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### Constants

e Euler's number: 2.718281828459
pi π, Pi: 3.1415926535898
pi2 π/2, Pi/2: 1.5707963267949
sq2 Square root of 2: 1.4142135623731
go Relation of the golden ratio: 1.6180339887499
d Feigenbaum constant delta: 4.6692016091030

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### Functions

Nested functions like sin(pow(x#2/3)) or polynomials like 2*x^3-4*x^2+x+1 are no problem at all. Functions with multiple variables, like norm, can have the x at any one or at several positions. The standard position is shown in the examples.

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### - Basic functions

^ or pow Power, e.g. x^2 or pow(x#2) for x2. Root can be written as e.g. x^(1/2) or x^.5 for square root of x, an exponential function like this: e^x for ex.
Roots of negative values can only be shown, if the numerator of the power is 1 and the denominator of the power is odd (e.g. x^(1/3) ). To calculate negative x-values for e.g. x^(2/3) , you have to alter this function into (x^(1/3))^2
sqr Square root, e.g. sqr(x) as equivalent to x^(1/2).
rt Root, e.g. third root rt(3#x) as equivalent to x^(1/3).
exp Exponent, e.g. exp(x) as equivalent to e^x.
ln or log Natural logarithm , e.g. log(x)
logn Logarithm to the base n, e.g. logn(2#x) for the binary (base 2) logarithm.

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### - Trigonometric and hyperbolic functions

 sin Sine, sinus, e.g. sin(x) cos Cosine, cosinus, e.g. cos(x) tan Tangent, e.g. tan(x) cot Cotangent, e.g. cot(x) sec Secant, e.g. sec(x) cosec Cosecant, e.g. cosec(x) sin2 Sine square, e.g. sin2(x) cos2 Cosine square, e.g. cos2(x) tan2 Tangent square, e.g. tan2(x) cot2 Cotangent square, e.g. cot2(x) sec2 Secant square, e.g. sec2(x) cosec2 Cosecant square, e.g. cosec2(x) arcsin Arcsine, e.g. arcsin(x) arccos Arccosine, e.g. arccos(x) arctan Arctangent, e.g. arctan(x) arccot Arccotangent, e.g. arccot(x) arcsec Arcsecant, e.g. arcsec(x) arccosec Arccosecant, e.g. arccosec(x) sinh Hyperbolic Sine, e.g. sinh(x) cosh Hyperbolic Cosine, e.g. cosh(x) tanh Hyperbolic Tangent, e.g. tanh(x) coth Hyperbolic Cotangent, e.g. coth(x) arsinh Area Hyperbolic Sine, e.g. arsinh(x) arcosh Area Hyperbolic Cosine, e.g. arcosh(x) artanh Area Hyperbolic Tangent, e.g. artanh(x) arcoth Area Hyperbolic Cotangent, e.g. arcoth(x) sech Hyperbolic Secant, e.g. sech(x) cosech Hyperbolic Cosecant, e.g. cosech(x) arsech Area Hyperbolic Secant, e.g. arsech(x) arcosech Area Hyperbolic Cosecant, e.g. arcosech(x)

cat Catenary, e.g. cat(2#x) for 2*cosh(x/2). The first value is the constant a.
gd Gudermannian function, e.g. gd(x) for arctan(sinh(x))
siv Semiversus, e.g. siv(x) for sin2(x/2)
sinc Sine cardinalis, e.g. sinc(x) for sin(x)/x
tanc Tanc function (not tangent cardinalis!), e.g. tanc(x) for tan(x)/x
hubb Hubbert curve, e.g. hubb(x) for 1/(2+2*cosh(x))
L Langevin function, e.g. L(x) for coth(x)-1/x

deg Converts the radian number to the equivalent number in degrees, e.g. deg(pi)

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### - Non-differentiable functions

abs Absolute value, e.g. abs(x)
min Minimum of several values, e.g. min(1#x#x^(1/3)) as minimum of 1, x and third root of x.
max Maximum of several values, e.g. max(abs(x)#x*x) as maximum of the absolute value of x and x2.
% Modulo division, whole-numbered remainder, e.g. 10%x
fmod Modulo division, floating point remainder, e.g. fmod(x#1) displays only the position after the decimal point of the input value.
R Round, e.g. R(x#2) rounds two decimal places, R(x) rounds to an integer.
R0 Floor (rounding down), e.g. R0(x)
R1 Ceil (rounding up), e.g. R1(x)
dist Distance function, e.g. dist(x) gives the distance to the nearest integer.
prime Prime number function, e.g. prime(x) This returns the next lower prime number (or x itself, if prime) for all x≥2 and x≤100000. At all four prime functions, non-integers are rounded.
prime1 Prime number detecting function, e.g. prime1(x) Displays a number only if prime, else 0. To find all prime numbers in an interval, the span of the x-axis shouldn't be wider than the width of the image (usually 500) and you should switch off poles.
prime2 Distinct prime factor counting function, e.g. prime2(x) returns the amount of different prime factors for an integer.
prime3 Prime factor counting function, e.g. prime3(x) returns the amount of prime factors for an integer, including multiples. E.g. prime2(4) = 1, whereas prime3(4) = 2. If prime3(x) = 1, then x is prime.
div Divisor function, e.g. div(x) returns the number of divisors of an integer. Non-integers are rounded.
dig Digit sum, e.g. dig(x) returns the digital sum of an integer. Non-integers are rounded, - is ignored.
dig2 Iterated (one-digit) digit sum, e.g. dig2(x) returns the iterated digital sum of an integer.
adig Alternating digit sum, e.g. adig(x) Non-integers are rounded, - is ignored.
fac Factorial, e.g. fac(x) Non-integers are rounded.
H Heaviside step function, e.g. H(x) 0, if x≤0, else 1.
Hm Multivariate Heaviside step function, e.g. Hm(x*x-1#sin(x)) 0, if at least one value ≤0, else 1. Enter as many arguments as you want.
sig Signum function (sign function), e.g. sig(x)
haar Haar wavelet, e.g. haar(x)
gcf Greatest common factor (or greatest common divisor, gcd), e.g. gcf(8#x) returns the greatest common factor between two integers. Non-integers are rounded.
lcm Least common multiple, e.g. lcm(8#x) returns the least common multiple between two integers. Non-integers are rounded.
mo Möbius function, e.g. mo(x) returns for all positive integers 0, if divisible by a square>1, -1 if it has an odd number of distinct prime factors and 1 if it has an even number of distinct prime factors. Non-integers are rounded. Values are allowed up to 100000.
toti Euler's totient function, e.g. toti(x) counts all positive integers less than x that are comprime to x. Non-integers are rounded.
odd Find odd numbers, e.g. odd(x) returns numbers only when odd. Non-integers are rounded.
even Find even numbers, e.g. even(x) returns numbers only when even. Non-integers are rounded.
bin Binomial coefficient, e.g. bin(4#x) The two values are n and k. Non-integers are rounded.
tri Triangle curve, e.g. tri(1#2#x) The first value is the period, the second is the amplitude.
rect Rectangle curve, e.g. rect(1#-1#2#x) The first value is the upper limit, the second is the lower and the third is the period.
saw Sawtooth wave, e.g. saw(2#1#x) The first value is the period, the second is the amplitude.
saw2 Reverse sawtooth wave, e.g. saw2(2#1#x) The first value is the period, the second is the amplitude.
ramp Ramp function, e.g. ramp(1#2#1#x) The first value is the start value, the second is the end value and the third is the height.
ramp2 Reverse ramp function, e.g. ramp2(1#2#1#x) The first value is the start value, the second is the end value and the third is the height.
trap Trapezium (trapezoid) function, e.g. trap(-4#-1#3#2#3#x) The first value is the start value of the climb, the second is the end value of the climb, the third is the height, the fourth is the start value of the descent and the fifth is the end value of the descent.
poly Polygon or chart line, e.g. poly(-4#2#-3#4#-2#1#-1#0#0#3#1#2#2#-1#3#3#4#1#x) gives a chart, respectively a half polygon. Here, (-4,2) is connected to (-3,4), this to (-2,1) and so on. The first value of each pair is the x-value, the second one is the y-value. The x-values must increase with each step. To get a full polygon, enter a second term with the same start and end points, like poly(-4#2#0.5#-4#4#1#x)
rand Integer random number between two integers, e.g. rand(0#2) returns 0, 1 or 2 (Mersenne twister is used for generating).
rand2 Random number between two numbers with decimal places (maximal 9), e.g. rand2(0#1#3) returns a number with three decimal places between 0 and 1 (Mersenne twister, too).

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### - Probability functions and statistics

norm Normal or Gaussian distribution, e.g. norm(0#1#x) for the standard normal distribution. The first value is the expected value, the second is the standard deviation.
phi Φ, Cumulative Gaussian distribution function, e.g. phi(0#1#x) This is an approximation based on the displayed interval. It delivers reasonable values, if the normal distribution in the chosen interval starts at very low values near 0. A common display of both functions is advisable.
lnorm Log-normal distribution, e.g. lnorm(0#1#x) The first value is the mean, the second is the standard deviation.
cau Cauchy distribution or Lorentz distribution, e.g. cau(0#1#x) for the standard Cauchy distribution. The first value is the location parameter, the second is the scale parameter.
lapc Laplace distribution, e.g. lapc(0#1#x) The first value is the location parameter, the second is the scale parameter. The second parameter must be >0.
logd Logistic distribution, e.g. logd(1#2#x) The first value is the location parameter, the second is the scale parameter.
hlogd Half-logistic distribution, e.g. hlogd(x)
rlng Erlang distribution, e.g. rlng(5#1#x) The first value is the shape parameter, the second is the rate parameter. The first parameter must be a natural number.
pon Exponential distribution, e.g. pon(1#x) The first value is the rate parameter.
cosd Raised cosine distribution, e.g. cosd(0#1#x) The first value is the location parameter, the second is the scale parameter. cosd is defined in the interval [location-scale;location+scale].
sechd Hyperbolic secant distribution, e.g. sechd(x)
kum Kumaraswamy distribution, e.g. kum(2#3#x) The first two values are the shape parameters a and b.
levy Lévy distribution, e.g. levy(1#x) The first value is the scale parameter.
rlgh Rayleigh distribution, e.g. rlgh(1#x) The first value is the scale parameter.
wb Weibull distribution, e.g. wb(2#1#x) The first value is the shape parameter, the second is the scale parameter.
wig Wigner semicircle distribution, e.g. wig(1#x) The first value gives the radius.
igauss Inverse Gaussian distribution, e.g. igauss(1#0.25#x) The first value is the shape parameter, the second is the scale parameter.
par Pareto distribution, e.g. par(2#1#x) The first value is the location parameter, the second is the shape parameter.
shg Shifted Gompertz distribution, e.g. shg(0.5#1#x) The first value is the scale parameter, the second is the shape parameter, both must be >0.
brw Relativistic Breit-Wigner distribution, e.g. brw(1#2#x) The first value is the mass of the resonance, the second is the resonance's width and the third is the energy.
gen Generalized extreme value distribution, e.g. gen(0#1#0.2#x) The first value is the location parameter, the second is the scale parameter and the third is the shape parameter.
Ft Fisher-Tippett distribution, e.g. Ft(1#2#x) The first value is the location parameter, the second is the scale parameter. The second parameter must be >0.
rossi Rossi distribution, or mixed extreme value distribution, e.g. rossi(0#3#1#4#x) The first four values are c1, c2, d1 and d2.
gum1 Gumbel distribution type 1, e.g. gum1(2#1#x) The first two values are the parameters a and b.
gum2 Gumbel distribution type 2, e.g. gum2(2#1#x) The first two values are the parameters a and b.
trid Triangular distribution, e.g. trid(1#2#4#x) The first value is the lower limit, the second is the most probable and the third is the upper limit.

#### - Discrete distributions

bind Binomial distribution, e.g. bind(5#0.4#x) The first value is the number of trials, the second is the success probability.
poi Poisson distribution, e.g. poi(3#x) The first value is λ, the second is the expected value.
skel Skellam distribution, e.g. skel(1#2#x) The first two values are the means of two different Poisson distributions.
gk Gauss-Kuzmin distribution, e.g. gk(x)
geo Geometric distribution (variant A), e.g. geo(0.8#x) The first value is a probability.
hgeo Hypergeometric distribution, e.g. hgeo(8#3#2#x) The first value is the total number of objects, the second is the total number of defective objects, the third is is the number of sample objects and the fourth the number of defective objects in the sample.
logs Logarithmic series distribution, e.g. logs(0.1#x) The first value is a probability.
zm Zipf-Mandelbrot law or Pareto-Zipf law, e.g. zm(100#1#2#x) The first three values are N, q and s. Maximum for N is 100.
uni Uniform distribution, e.g. uni(1#2#x) The first value is the lower limit, the second is the upper limit.

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### - Special functions

traj Trajectory parabola, path of a thrown object, e.g. traj(45#20#9.81#x) The first value is the angle, the second is the speed (e.g. in meters per second). The third value is the gravitational acceleration (e.g. in m/s²), the normal value on earth for this is g = 9.81 m/s². The axes scale in this example is meters. Air resistance is ignored.
pll Parallel operator, as used, among others, for the calculation of parallel circuits and resistances, e.g. pll(20#30#x) Enter as many arguments as you want.
M1 Arithmetic mean, e.g. M1(2#3#x) Enter as many arguments as you want.
M2 Geometric mean, e.g. M2(2#3#x) Enter as many arguments as you want, only positive values are allowed.
M3 Harmonic mean, e.g. M3(2#3#x) Enter as many arguments as you want, only positive values are allowed.
M4 Root mean square, e.g. M4(2#3#x) Enter as many arguments as you want.
M5 Median, e.g. M5(2#3#x) Enter as many arguments as you want.
scir Semicircle curve, e.g. scir(x#1) for a semicircle with the radius 1. The formula is sqr(r*r-x*x), r gives the radius.
ell Semielliptic curve, e.g. ell(2#1#x) for a semiellipse with the horizontal radius 2 and the vertical radius 1. The formula is sqr((1-x*x/(a*a))*b*b).
ell2 Semi-superellipse or semi-hyperellipse, e.g. ell2(2#3#4#x) for a semiellipse with the horizontal radius 2, the vertical radius 3 and n=4.
lmn Lemniscate of Bernoulli, e.g. lmn(1#x) This returns a half lemniscate. For the other half, use -lmn(1#x)
lmn2 Lemniscate of Gerono, e.g. lmn2(x) This returns a half lemniscate. For the other half, use -lmn2(x)
lmn3 Lemniscate of Booth, e.g. lmn3(1#x) This returns a half lemniscate. For the other half, use -lmn3(1#x)
pyth Pythagorean theorem, e.g. pyth(x#1) The formula is c=sqr(a*a+b*b).
thr Rule of three, e.g. thr(x#1#2) The formula for thr(a#b#c) is f(x)=b*c/a.
fib Fibonacci numbers, e.g. fib(x) or fib(x#1) If the second value is 1, a continuous graph is shown, else a discrete.
dc Exponential decay, e.g. dc(5#1#x) The first value is the initial quantity, the second is the decay constant.
erf Gaussian error function, e.g. erf(x) For the computation its Taylor series is used.
HY4 Hyper4, also known as tetration or super-exponentiation, e.g. HY4(x#3) for x to the power of (x to the power of x). Here the maximum value can be excessed very quickly!
lambda Lambda function, e.g. lambda(x#3) for x to the power of (x to the power of (3-1)).
sgm Sigmoid function, e.g. sgm(x) for 1/(1+e^(-x)).
gom Gompertz curve, e.g. gom(2#-5#-3#x) The first value is the upper asymptote, the second is the parameter b and the third is the growth rate. Second and third value must be negative.
stir Stirling's approximation for large factorials, e.g. stir(x) The formula is (2*pi*x)^(1/2)*(x/e)^x.
omega Lambert-W function or Omega function or product log (approximation), e.g. omega(x)
bump Bump function psi, ψ, e.g. bump(x) for exp(-1/(1-x*x)) between -1 and 1, else 0.
srp Serpentine curve, e.g. srp(2#1#x) The formula is a*a*x/(x*x+a*b). The first two values are a and b.
bsc Gaussian bell-shaped curve, e.g. bsc(1#x) The formula is exp(-a*a*x*x), the first value is the shape parameter a.
gbsc Generalized Gaussian bell-shaped curve, e.g. gbsc(1#2#-1#x) for 1*exp(2*x-1*x*x).

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### - Programmable functions

bool Characteristic boolean function, e.g. bool(1/x) Returns nothing, if the input value is not defined, 0, if 0, else 1.
bool0 Defined boolean function, e.g. bool0(x) Returns 0, if the input value is 0 or not defined, else 1.
bool1 Undefined boolean function, e.g. bool1(prime1(x)) Returns nothing, if the input value is 0 or not defined, else 1.
con Condition function, e.g. con(0#sin(x)#1) The first value is the lower limit, the third is the upper limit. If the second value is between these two, the result is 1, else 0.
rcon Reverse condition function, e.g. rcon(0#sin(x)#1) The first value is the lower limit, the third is the upper limit. If the second value is between these two, the result is 0, else 1.
wcon Weighted condition function, e.g. wcon(0#sin(x)#1) Only returns the second value, if this lies between the first and the third value.
rwcon Reverse weighted condition function, e.g. rwcon(0#sin(x)#1) Only returns the second value, if this doesn't lie between the first and the third value.
&& (and) can be simulated with the minimum function, e.g. min{ con[0#sin(x)#1] # con[0#cos(x)#1] }
|| (or) can be simulated with the maximum function, e.g. max{ con[0#sin(x)#1] # con[0#cos(x)#1] }
⊕ (xor) can be simulated with the maximum minus the minimum function, e.g.
max{ con[0#sin(x)#1] # con[0#cos(x)#1] } - min{ con[0#sin(x)#1] # con[0#cos(x)#1] }

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### - Iterations (iterative functions)

y Previous function value, e.g. for y(0)+0.01 is 0 the initial value for y, the next value is the last result of the input value x and so on.
y2 Pre-previous function value, e.g. y2(1)+0.001
step Number of the iteration steps done, divided by the parameter value, e.g. step(100) counts up to five (at 500 px width).
mean Iterated arithmetic mean, e.g. mean(sin(x)) gives the arithmetic mean of the y-values returned to the so far reached x-values.
man Mandelbrot function, e.g. man(0#-1.9) for y(0)*y(0)-1.9.
Attention: derivative and integral with the iteration don't lead to very reasonable results. As well a logarithmic scale won't work here.

#### - Fractals

rsf Random singular function (a kind of devil's staircase), e.g. rsf(0#2) for y(a)+0.008*rand(0#1)*rand(0#1)*(b-a), from a (first value) to b (second value) at 500px width. The first value is the start point on the y-axis, the second is the average end value.
wf Weierstrass function, e.g. wf(x#0.5#17#10) The second value is a parameter between 0 and 1, the third value is a positive, odd integer. The second multiplied with the third must be larger than 1+3/2*pi. The fourth value is the number of steps done. In theory this is infinite, but here the maximum is 100.
blanc Blancmange curve, e.g. blanc(x#10) The second value is the number of steps done, maximum is 1000.
tak Takagi-Landsberg curve, e.g. tak(x#0.7#10) The second value is a parameter, which should be between 0 and 1. The third is the number of steps done, maximum is 1000.

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### Differential and integral equations

Derivatives within a function are written like this:
 D or D1 First derivative, e.g. D(x*x) D2 Second derivative, e.g. D2(x^3) D3 Third derivative, e.g. D3(x^4) D0 or D01 First derivative, alternative form, e.g. D0(x*x) D02 Second derivative, alternative form, e.g. D02(x^3) D03 Third derivative, alternative form, e.g. D03(x^4)
Only one derivative of each form can be included in one function. So something like D(...)+D2(...) or D(...)+D0(...) will work, D(...)+D(...) won't, D(...)+D1(...) won't either.
Derivative of fourth order over a whole function can be generated by using D3 together with selecting 'Derivative' in the display or by writing e.g. D(D3(sin(x))) (not D(D(...))). The maximum for reasonable results is seventh order, e.g. D(D3(D03(...))).

Integrals within a function are written like this:
S or S1 First integral, e.g. S(x*x)
S2 Second integral, e.g. S2(x)
S3 Third integral, e.g. S3(1)
Only one integral can be included in one function. So something like S(...)+S2(...) won't work.
Integral of fourth order over a whole function can be generated by using S3 together with selecting 'Integral' in the display.

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Just try out, you can't break anything!

You can pick Transp. (transparent) at every place where you can choose a color. When selecting it for any of the three function graphs or the background, it is better to switch antialiasing off. Transparency can only be displayed in png and gif images, not in jpeg.

### Functions:

Hull is a function, which is laid above the three functions f(x), g(x) and h(x), using Y as a replacement for them. E.g. sin(Y) gives back the sine for each defined function of the three. This can also be used to easily make a similar damping for oscillations, e.g. by using 2*exp(-.5*x)*Y as hull, sin(x) as f(x) and sin2(x) as g(x).

Next to the formula terms, the color of up to three graphs can be set and whether the according term should be written into the graphic. You can also choose, if the dots should be connected or not, or if the in- or outside area of the graph should be filled.
Derivative displays the derivated graph. In the graphic, this will be displayed as f'(x)=[...]'.
Integral you can choose to display the cumulative function in the displayed interval (integrate over ...). Thereby the function values are cumulated one after another. In the graphic, the integrated term will be displayed as F(x)=S[...]. You can also set a constant C, which will be added to the integral.
At From ... to, you can choose the domain for the function (piecewise defined function). Enter the designated x-values. If empty, the domain will cover the whole range for x. Constants like pi are allowed, too.

### Display properties:

At Image type you can choose png (compression level 1), gif (GIF87a) or jpeg (quality level 90 percent). png is recommended.
Width and Height refer to the size of the graphic and have nothing to do with the range of values (except at iterations). Minimum size is 200, maximum is 500.
Range defines, in which range the graphs are displayed. Maximal input and output value is 100000 (or -100000). With a logarithmic scale, the output value can raise up to about 10300. Constants like pi are allowed, too.
Intervals defines the number of sectors on each axis that are labeled with dashes and numbers. Maximum is 250 or half of the width / height. The width should be divisible without remainder by the number of intervals on the x-axis, same for the height and the intervals on the y-axis. But this is no restraint.
Reticule lines defines the number of the drawn through lines (or grid). Maximum is 250 or half of the width / height.
Dashes length defines the length of the dashes at the interval borders. Maximum length is 500.
Decimal places defines the number of displayed decimal places in the caption and for the calculation of single values. Maximum is 12.
Gap at origin the size of a gap around the origin. When 0, this isn't shown, for 1 is makes a square through x=-1, x=1, y=-1 and y=1, in which only the graph is shown.
Circle at origin the size of a circle around the origin. When 0, this isn't shown, for 1 is makes a circle through x=-1, x=1, y=-1 and y=1. When the range of values of the x- and y-axis doesn't have the same span, the circle will become an ellipse.
Graph thickness determines the thickness of the function graphs. Values can be positive integers up to 200.

Log. scale defines, if the x- and y-axis is displayed linear or logarithmic. No means linear. As logarithmic base you can choose 2, e, 10 or 100, or enter an individual value. The logarithmic display doesn't show integrated or derivated graphs or iterations!

Quadrants offers several buttons to change the displayed quadrants and an input field for their size. If you wish to change the size, do that before you click on one of these buttons.

Mark points at finds points on the graph. Enter different values separated by spaces for each graph. E.g. 1 2 marks the points (1;f(1)) and (2;f(2)). Not at derivative and integral.

The checks, if set, cause the display of the reticule lines, axes lines (x- and y-axis), caption (values and axes), dashes, frame and potentially occuring error messages.

At Def. Q= you can define a formula or a constant and then use Q for a substitute in the three formula terms. When inserted, the Q term will be put in parentheses, to avoid unexpected results. E.g. 1+x for Q will become (1+x).

In the next line you can define the colors of the display and choose Antialiasing (which often looks better). Poles, if selected, finds poles and doesn't connect them. Unfortunately, this feature is far from being perfect.
You can also decide, whether lines and caption should be drawn in the background or in the foreground or not at all.
At Gamma, you can make a gamma correction, insert a value larger than 0 here, 1 is standard.
At Brightness, insert an integer between -255 and 255, 0 doesn't change the brightness.
At Contrast, insert an integer between -100 and 100, 0 doesn't change anything.
At Rotation, enter the angle in degrees, which you wish the image to rotate.
Emboss, Blur, Negative, Greyscale, Sketchy and Edges only are special effects which, when checked, will change the look of the graph as their name implies.
Self-defined color lets you define three colors of your own choice, which can be chosen as Self 1-3 in the color lists. Enter the color as a 6-digit hex code, e.g. ffffff for white. Color calculator to find out color codes.

The displayed graph is never an exact image of the function, but an approximation as good as possible.

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# Calculate single value

Enter a function with the syntax above or click on 1, 2 or 3 to take a specified function term. Choose a numeric (no formula) input value and click Calculate to see its function value. The result is rounded according to the setting at Decimal places. Values of the function phi, derivatives, integrals and iterations can't be calculated. The substitution Q will be calculated, the hull Y won't.
Enter several values, separated by spaces (e.g. 1 2 3 4 5), to get a score table. Press +10 to get the numbers from 1 to 10, or -10 to get those from -1 to -10. You can also choose, if you want as output only the Results, a Table with the x-values as well or a CSV-type list (semicolon separated).
This tool can also be used as a pocket calculator. Simply enter an arithmetic term like 2*2 and no input value.

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To save a setting, copy the path text in the input field and save it in a text file. You can later paste it into this field and load it again.

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# Gimmicks & Fun

You can as well use this plotter for doing geometrical graphic designs like for logos and such. Therefore especially the functions for filling the graph out (or in) are useful, combined with a background color. Although you are restricted to the mathematical functions above, you will be surprised what is possible to create with a little bit of effort and practice.

Examples:
To view the examples, copy the code in the Load & Save section (clear it before) and click on Load Graph.

### Starry Sky

Set the background to black and remove all lines and caption. Use the term rand2(-5#5#3), choose a starlike color for it (yellow, red, even blue) and draw it dotted. If you want more stars, use this term for all three graphs. If you want less, use something like rand2(-10#10#3) instead. In the following code I have also changed brightness and contrast and reduced the number of blue stars. Each time you draw this image new, you'll get a different sight.
a0=2&a1=Q&a2=Q&a3=Q*5&a4=5&a5=3&a6=0&a7=&a8=&a9=&b0=500&b1=500&b2=-5&b3=5&b4=-5&b5=5&b6=10&b7=10&b8=5&b9=5&c0=3&c1=0&c2=1&c3=1&c4=1&c5=&c6=1&c7=0&c8=0&c9=0&d0=1&d1=20&d2=20&d3=0&d4=&d5=&d6=&d7=&d8=&d9=&e0=&e1=&e2=&e3=rand2(-5#5#3)&e4=13&e5=14&e6=13&e7=12&e8=1&e9=1&f0=1&f1=1&f2=1&f3=-80&f4=-50&f5=&f6=&f7=&f8=&f9=&g0=&g1=0&g2=1&g3=0&g4=0&g5=0&g6=Y&g7=ffffff&g8=ffffff&g9=ffffff&h0=1&h1=&h2=&h3=&h4=0&z

### Kind of a Star Trek logo

Take -pow(x#2)+9 as the first and -pow(x#2)+5 as the second term. The y-axis has to run from 0 to 10. Again, remove all lines and captions and make the background black. For the first graph, choose a light color, for the second black. Set both to fill in.
a0=2&a1=-pow(x#2)+9&a2=-pow(x#2)+5&a3=&a4=5&a5=13&a6=8&a7=&a8=&a9=1&b0=500&b1=500&b2=-5&b3=5&b4=0&b5=10&b6=10&b7=10&b8=5&b9=5&c0=3&c1=0&c2=1&c3=1&c4=1&c5=1&c6=1&c7=0&c8=0&c9=0&d0=1&d1=20&d2=20&d3=0&d4=&d5=&d6=&d7=&d8=&d9=&e0=&e1=&e2=&e3=&e4=13&e5=14&e6=13&e7=12&e8=2&e9=2&f0=0&f1=1&f2=1&f3=0&f4=0&f5=&f6=1&f7=&f8=&f9=&g0=&g1=0&g2=1&g3=0&g4=0&g5=0&g6=Y&g7=ffffff&g8=ffffff&g9=ffffff&h0=1&h1=&h2=&h3=&h4=0&z

### Plaid Cyclone

Just had a bit of fun with this one.
a0=2&a1=cat(2*(sqr(abs(x))-x/8)#x)&a2=6&a3=rand2(-2#0#2)&a4=1&a5=1&a6=9&a7=&a8=&a9=&b0=500&b1=500&b2=-5&b3=5&b4=-2&b5=8&b6=10&b7=10&b8=5&b9=5&c0=3&c1=0&c2=&c3=&c4=&c5=1&c6=1&c7=0&c8=0&c9=0&d0=1&d1=50&d2=50&d3=0&d4=&d5=&d6=&d7=&d8=&d9=&e0=&e1=&e2=&e3=&e4=18&e5=6&e6=13&e7=1&e8=2&e9=3&f0=0&f1=1&f2=1&f3=0&f4=0&f5=&f6=&f7=&f8=&f9=&g0=&g1=1&g2=1&g3=0&g4=0&g5=0&g6=Y&g7=ffffff&g8=ffffff&g9=ffffff&h0=1&h1=&h2=&h3=&h4=0&z

### Propeller

a0=2&a1=x&a2=abs(x)^.9*sig(x)&a3=-(x/25)^3&a4=17&a5=37&a6=17&a7=&a8=&a9=&b0=500&b1=500&b2=-1000&b3=1000&b4=-1000&b5=1000&b6=10&b7=10&b8=5&b9=5&c0=3&c1=30&c2=&c3=&c4=&c5=&c6=&c7=0&c8=0&c9=0&d0=1&d1=10&d2=10&d3=0&d4=&d5=&d6=&d7=&d8=-120&d9=120&e0=&e1=&e2=&e3=&e4=37&e5=13&e6=13&e7=37&e8=2&e9=2&f0=2&f1=1&f2=1&f3=0&f4=0&f5=&f6=&f7=&f8=&f9=&g0=&g1=2&g2=1&g3=90&g4=0&g5=0&g6=Y&g7=ffffff&g8=ffffff&g9=ffffff&h0=1&h1=&h2=&h3=&h4=0&z

### Strange object

a0=2&a1=2^x&a2=x^2&a3=-200*x+3000&a4=6&a5=13&a6=13&a7=&a8=&a9=&b0=400&b1=400&b2=4&b3=15&b4=20&b5=1010&b6=10&b7=10&b8=5&b9=5&c0=3&c1=0&c2=&c3=&c4=&c5=&c6=&c7=0&c8=0&c9=0&d0=1&d1=20&d2=20&d3=0&d4=&d5=&d6=&d7=&d8=&d9=&e0=&e1=&e2=&e3=&e4=13&e5=14&e6=11&e7=11&e8=2&e9=2&f0=3&f1=&f2=1&f3=0&f4=0&f5=&f6=1&f7=&f8=&f9=&g0=&g1=1&g2=1&g3=35&g4=0&g5=0&g6=Y&g7=ffffff&g8=ffffff&g9=ffffff&h0=1&h1=&h2=&h3=&h4=0&z

### Prime number saw

a0=2&a1=prime(x)&a2=x-25&a3=x-300&a4=4&a5=3&a6=38&a7=&a8=&a9=&b0=500&b1=500&b2=500&b3=0&b4=0&b5=500&b6=10&b7=10&b8=5&b9=5&c0=3&c1=0&c2=1&c3=1&c4=1&c5=&c6=1&c7=0&c8=0&c9=0&d0=&d1=20&d2=20&d3=0&d4=&d5=&d6=&d7=&d8=&d9=&e0=&e1=&e2=&e3=&e4=38&e5=14&e6=21&e7=1&e8=2&e9=2&f0=2&f1=1&f2=1&f3=0&f4=0&f5=&f6=&f7=&f8=&f9=&g0=&g1=0&g2=1&g3=140&g4=0&g5=0&g6=Y&g7=ffffff&g8=ffffff&g9=ffffff&h0=1&h1=&h2=&h3=&h4=0&z

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Something missing? Suggestions or criticism please mail to juergen [at] jumk [dot] de

Due to overload issues, the plotter can't display extremely computational intensive functions, like the gamma functions and those which use this. Those can be found in the source code.

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Have fun ;-)

Open source code: openPlaG | Gallery