Function grapher manual

Constants
Functions
Complex functions
Operators
Syntax and semantics
The graphers in detail
Examples
Advanced

Constants

e......Euler's number
pi.....Circle constant
tau....2*pi
deg....pi/180
gon....pi/200
gc.....Euler-Mascheroni constant
gr.....golden ratio

Functions

Basic functions
abs(x)........absolute value
sgn(x)........signum function
sqrt(x).......square root
root(n,x).....nth root
floor(x)......rounds x down
ceil(x).......rounds x up
rd(x).........rounded value
frac(x).......fractional part
max(a,b)......maximum
min(a,b)......minimum
mod(x,m)......modulo funktion
H(x)..........unit step function

exp(x)........natural exponential function
ln(x).........natural logarithm
lg(x).........decadic logarithm
ld(x).........logarithm to base two
log(x,b)......logarithm to base b

Trigonometric functions
sin(x)........sine
cos(x)........cosine
tan(x)........tangent
cot(x)........cotangent
sec(x)........secant
csc(x)........cosecant

asin(x).......arc sine
acos(x).......arc cosine
atan(x).......arc tangent
acot(x).......arc cotangent
asec(x).......arc secant
acsc(x).......arc cosecant

gd(x).........Gudermann function
hypot(x,y)....length of the coordinate vector (x,y)
angle(x,y)....angle of the coordinate vector (x,y)

Hyperbolic functions
sinh(x).......sine hyperbolicus
cosh(x).......cosine hyperbolicus
tanh(x).......tangent hyperbolicus
coth(x).......cotangent hyperbolicus
sech(x).......secant hyperbolicus
csch(x).......cosecant hyperbolicus

asinh(x)......area sine hyperbolicus
acosh(x)......area cosine hyperbolicus
atanh(x)......area tangent hyperbolicus
acoth(x)......area cotangent hyperbolicus
asech(x)......area secant hyperbolicus
acsch(x)......area cosecant hyperbolicus

Wave forms
twave(x,p)....triangle wave, p-periodic
ptwave(x,p)...positive triangle wave, p-periodic
sqwave(x,p)...square wave, p-periodic
psqwave(x,p)..positive square wave, p-periodic
stwave(x,p)...sawtooth wave, p-periodic
pstwave(x,p)..positive sawtooth wave, p-periodic

Analysis
diff(f,x).....derivative of f at x
diff(f,x,2)...second derivative of f at x
diff(f,x,n)...nth derivative of f at x
int(f,0,x)....integral(from 0 to x) f(t) dt
int(f,0,x,n)..integration with n steps
pow(f,n,x)....composition of function f with itself
sum(f,a,b)....sum(k=a to b) f(k)
prod(f,a,b)...product(k=a to b) f(k)
rand(a,b).....random number in the interval [a,b)
inv(f,x)......inverse function f^(-1)(x)
inv(f,x,a,b)..f^(-1)(x) for f defined on [a,b]

Enhancement functions
[a1,...,an]...constructs a list from the values a1,...,an
get(a,i)......element number i of the list a
size(a).......number of elements in the list a
cat(a,b)......concatenates the lists a and b
range(a,b)....constructs the list [a,a+1,a+2,...,b]
range(a,b,d)..constructs the list [a,a+d,a+2d,...,b]
next(a).......cyclically picks an element from the list a
rand(a).......randomly picks an element from the list a
sum(a)........sum of all elements in the list a
prod(a).......product of all elements in the list a
max(a)........maximum of all elements in the list a
min(a)........minimum of all elements in the list a
rev(a)........reversed list
ipp(a)........interpolation polynomial, a=[[x0,y0],...,[xn,yn]]
fn(a).........piecewise linear interpolation
fn(x0,d,a)....piecewise linear interpolation, equidistant nodes
map(f,a)......applys the function f to all elements in a
filter(f,a)...filters elementes from the list a
reduce(f,a)...reduces a with f(x,y)
compose(a)....composition of all functions in the list a
count(f,a)....counts how often f(x) for x in a is true
forall(f,a)...f(x) is true for all x in a
exists(f,a)...f(x) is true for any x in a
table(f,a)....lookup table of f for the list a
apply(f,x)....application of f to x
if(c,a,b).....if c then a else b
not(a)........logical negation
clamp(x,a,b)..min(max(x,a),b)
sma(f,x,n,h)..central simple moving average

Gamma and related
fac(x)........factorial
gamma(x)......gamma function
lngamma(x)....ln(gamma(x)) for large x
digamma(x)....digamma function
pgamma(n,x)...polygamma function
B(a,b)........beta function
gamma(s,x)....lower incomplete gamma function
Gamma(s,x)....upper incomplete gamma function
B(x,a,b)......incomplete beta function B[x](a,b)
I(x,a,b)......B(x,a,b)/B(a,b)
G(x)..........Barnes G-function
hyperK(x).....K-function

Combinatorical functions
bc(n,k).......binomial coefficient
ff(n,k).......falling factorial
rf(n,k).......raising factorial
s1(n,k).......Stirling number of the first kind
s2(n,k).......Stirling number of the second kind

Elliptic integrals and related
K(m)..........complete elliptic integral K(m=k^2)
E(m)..........complete elliptic integral E(m=k^2)
F(phi,m)......incomplete elliptic integral F(phi,m=k^2)
E(phi,m)......incomplete elliptic integral E(phi,m=k^2)
Pi(phi,n,m)...incomplete elliptic integral Pi(phi,n,m=k^2)
RF(x,y,z).....Carlson symmetric form RF(x,y,z)
RJ(x,y,z,p)...Carlson symmetric form RJ(x,y,z,p)
RC(x,y).......shorthand for RF(x,y,y)
RD(x,y,z).....shorthand for RJ(x,y,z,z)
agm(a,b)......arithmetic-geometric mean
magm(a,b).....modified arithmetic-geometric mean

Antiderivatives
Ei(x).........exponential integral
En(n,x).......exponential integral E[n](x)
li(x).........integral logarithm
Li(x).........integral logarithm with offset li(2)
Si(x).........sine integral
Ci(x).........cosine integral

Bessel and related
BJ(a,x).......Bessel function J[a](x)
BY(a,x).......Bessel function Y[a](x)
Bj(a,x).......spherical Bessel function j[a](x)
By(a,x).......spherical Bessel function y[a](x)
BI(a,x).......modified Bessel function I[a](x)
BK(a,x).......modified Bessel function K[a](x)
Ai(x).........Airy function Ai(x)
Bi(x).........Airy function Bi(x)

Hypergeometric functions
F([a1,...,am],[b1,...,bn],x)
..............hypergeometric function
M(a,b,x)......confluent hypergeometric function M(a,b,x)
U(a,b,x)......confluent hypergeometric function U(a,b,x)

Zeta and related
zeta(s).......zeta function
zeta(s,a).....Hurzwitz zeta function
Phi(x,s,a)....Lerch transcendent
Li(s,x).......polylogarithm Li[s](x)
B(k)..........Bernoulli number, B(1)=+1/2
Bm(k).........Bernoulli number, Bm(1)=-1/2

Polynomial functions
PP(n,a,x).....Legendre polynomial P[n,a](x)
PL(n,a,x).....Laguerre polynomial L[n,a](x)
PH(n,x).......Hermite polynomial H[n](x)
PT(n,x).......Chebyshev polynomial T[n](x)
PU(n,x).......Chebyshev polynomial U[n](x)

Fourier analysis
fa(f,i,j).....Fourier coefficients a[i] upto a[j]
fb(f,i,j).....Fourier-coefficients b[i] upto b[j]
fs(x,a0,a,b)..Fourier series, a=[a1,a2,...], b=[b1,b2,...]
fs(x,a0,a)....amplitude-phase form(cos), a=[[A1,phi1],[A2,phi2],...]
Fs(f,x).......sine transform, x in Hz
Fc(f,x).......cosine transform, x in Hz
comb(x,a,p)...Dirac comb of sharpness a und period p

Functional analysis
sinc(x).......cardinal sine: sin(pi*x)/(pi*x)
delta(x,a)....Dirac delta function
L(f,x)........Laplace transform
E(a,b,x)......Mittag Leffler function
D(f,x,r)......fractional derivative (D^r)(f)(x)
J(f,a,x,r)....Riemann-Liouville integral (J[a]^r)(f)(x)
J2(f,a,x,r)...Riemann-Liouville integral, alternative algorithm

Other functions
W(x)..........Lambert W-function, upper branch
Wm1(x)........Lambert W-function, lower branch

Number theoretic functions
pm(x,n,m).....modular power
gcd(a,b)......greatest common divisor
lcm(a,b)......least common multiple
isprime(n)....prime number test
primes(m,n)...the list of prime numbers between m and n
factor(n).....prime number factorization
pcf(n)........prime number counting function pi(n)
phi(n)........Euler's totient function
lambda(n).....Carmichael function
sigma(n,k)....divisor function sigma[k](n)

Statistical functions
erf(x)........error function
norm(x).......cumulative standard normal distribution
mean(a).......arithmetic mean of the elements of a
sd(a).........standard deviation of the elements of a
cdf(a,x)......CDF for the random numbers in a
pmf(a,x)......PMF, a is a list of integer numbers

pmfB(n,p,k)...binomial distribution, PMF
cdfB(n,p,k)...binomial distribution, CDF
pmfG(p,k).....geometric distribution, PMF
cdfG(p,k).....geometric distribution, CDF
pmfH(N,K,n,k)...hypergeometric distribution, PMF
cdfH(N,K,n,k)...hypergeometric distribution, CDF
pmfP(lambda,k)...Poisson distribution, PMF
cdfP(lambda,k)...Poisson distribution, CDF
pmfLog(p,k)...logarithmic distribution, PMF
cdfLog(p,k)...logarithmic distribution, CDF

pdfN(mu,sigma,x)...normal distribution, PDF
cdfN(mu,sigma,x)...normal distribution, CDF
pdfLogN(mu,sigma,x)...log-normal distribution, PDF
cdfLogN(mu,sigma,x)...log-normal distribution, CDF
pdfExp(lambda,x)...exponential distribution, PDF, E=1/lambda
cdfExp(lambda,x)...exponential distribution, CDF, E=1/lambda
pdfst(n,x)....student's t-distribution, PDF
cdfst(n,x)....student's t-distribution, CDF
pdfF(m,n,x)...Fisher distribution, PDF
cdfF(m,n,x)...Fisher distribution, CDF
 qfF(m,n,p)...Fisher distribution, quantile function
pdfW(a,b,x)...Weibull distribution, PDF, E=b*gamma(1+1/a)
cdfW(a,b,x)...Weibull distribution, CDF, E=b*gamma(1+1/a)
pdfGamma(b,p,x)...Gamma distribution, PDF, E=p/b
cdfGamma(b,p,x)...Gamma distribution, CDF, E=p/b
pdfBeta(p,q,x)...Beta distribution, PDF
cdfBeta(p,q,x)...Beta distribution, CDF
pdfchisq(n,x)...chi-squared distribution, PDF
cdfchisq(n,x)...chi-squared distribution, CDF
 qfchisq(n,p)...chi-squared distribution, quantile function

Complex functions

Basic functions
re(z).........real part
im(z).........imaginary part
conj(z).......conjugation
arg(z)........phase
abs(z)........absolute value
sgn(z)........signum function
sqrt(z).......square root
root(n,z).....nth root
exp(z)........natural exponential function
ln(z).........natural logarithm
lg(z).........decadic logarithm
log(z,b)......logarithm to base b

Trigonometric functions
sin(z)........sine
cos(z)........cosine
tan(z)........tangent
cot(z)........cotangent

asin(z).......arc sine
acos(z).......arc cosine
atan(z).......arc tangent
acot(z).......arc cotangent

Hyperbolic functions
sinh(z).......sine hyperbolicus
cosh(z).......cosine hyperbolicus
tanh(z).......tangent hyperbolicus
coth(z).......cotangent hyperbolicus

asinh(z)......area sine hyperbolicus
acosh(z)......area cosine hyperbolicus
atanh(z)......area tangent hyperbolicus
acoth(z)......area cotangent hyperbolicus

Complex analysis
diff(f,z).....complex derivative
diff(f,z,n)...nth complex derivative
int(f,a,b)....integral(from a to b) f(t) dt
cint(f,g).....contour integral(from g(0) to g(1)) f(z) dz
vint(f,a).....contour integral(from get(a,1) to get(a,n)) f(z) dz

Gamma and related
fac(z)........factorial
gamma(z)......gamma function
digamma(z)....digamma function
gamma(s,z)....lower incomplete gamma function
Gamma(s,z)....upper incomplete gamma function

Elliptic integrals and related
K(m)..........complete elliptic integral K(m=k^2)
E(m)..........complete elliptic integral E(m=k^2)

Antiderivatives
erf(z)........error function
Ei(z).........exponential integral
En(n,z).......exponential integral E[n](z)

Zeta and related
zeta(s).......zeta function
zeta(s,a).....Hurzwitz zeta function
Phi(x,s,a)....Lerch transcendent
B(k)..........Bernoulli number

Other
theta1(z,q)...theta function theta1(z,q)
theta2(z,q)...theta function theta2(z,q)
theta3(z,q)...theta function theta3(z,q)
theta4(z,q)...theta function theta4(z,q)
sn(z,k).......sinus amplitudinis
cn(z,k).......cosinus amplitudinis
dn(z,k).......delta amplitudinis

Operators

01 f\x.....f applied to x
02 a^b.....a to the power of b
03 +a......plus a
03 -a......minus a
04 a*b.....a times b
04 a/b.....a divided by b
04 a%b.....a modulo b
05 a+b.....a plus b
05 a-b.....a minus b
06 a:b.....[a, a+1, a+2, ..., b]
06 a:b:d...[a, a+d, a+2d, ..., b]
07 a<b.....if a is less than b then 1 else 0
07 a>b.....if a is greater than b then 1 else 0
07 a<=b....if a is less than or equal to b then 1 else 0
07 a>=b....if a is greater than or equal to b then 1 else 0
08 a=b.....if abs(a-b)<epsilon then 1 else 0
08 a!=b....if abs(a-b)>epsilon then 1 else 0
09 a&b.....a and b
10 a|b.....a or b

Syntax and semantics

Care should be taken of the following:

* The decimal mark is a point.
  (the sign which separates fractional part from integer part)
* x to the power of 2 is written x^2.
* In place of x^-1 you have to write x^(-1).
* The power operator is left-associative.
  That means a^b^c = (a^b)^c.
* x/a/b means (x/a)/b.
* In place of 2*x you can also write 2x.
* In place of (x+2)*(x-2) you can also write (x+2)(x-2).
* But in place of x(x-1) you have to write x*(x-1).
* Omitting the multiplication sign in a*x and x*x is also
  not allowed.

If a hash sign (#) is written at the beginning of the input line,
then the following function is defined, but not graphed.
For example the follwing input is possible:
  f(x) = # sin(x)
  g(x) = f(2x)

The notation {x| term of x} is used for the definition of anonymous
functions. For example, the function which maps x to 2x is written
{x|2x}, or alternatively {t|2t} or {u|2u}. This function can be
applied to the variable x like a named function: {t|2t}(x). This is
advantageous in certain cases. For example, the function
  f(x) = 1+(x-2)+(x-2)^2/2+(x-2)^3/6
can also be written as
  f(x) = {x|1+x+x^2/2+x^3/6}(x-2).

The derivative of x^2 can be stated as
  f(x) = diff({t|t^2},x).
An alternative formulation is
  f(x) = x^2,
  g(x) = diff(f,x).

To draw f(x) = integral(from 0 to x) 2t dt,
write f(x) = int({t|2t},0,x).

The operator ":=" is used to assign a value to a variable.
Instead of
  f(x,y) = sin(sqrt(x^2+y^2))/sqrt(x^2+y^2)
you may also write
  f(x,y) = r:=sqrt(x^2+y^2), sin(r)/r.

It is also possible to draw a bunch of functions at the same time.
Examples:
  f(x) = next([x,x^2,1/x])
  f(x) = next([-2,-1,1,2])*sin(x)
  f(x) = next(-4:4)*sin(x)
If the graphs of the functions are not continuous,
to make them continous increase the number of points.

If a statement has to be evaluated only once, the statement has to
appear behind a semicolon. For example, there is a difference between
  f(x) = r:=rand(0,1), x+r
and
  f(x) = x+r; r:=rand(0,1).

The graphers in detail

Grapher for real functions
px: viewpoint, x-coordinate
py: viewpoint, y-coordinate
wx: width of the view
wy: height of the view

The x-interval of the view is
  [px-wx, px+wx]
and the y-interval is
  [py-wy, py+wy].

n: number of points in the interval
  [px, px+wx]

a: parameter
d: distance (a:=a-d, a:=a+d)
b: parameter
d: distance (b:=b-d, b:=b+d)

Grapher for implicit functions
To draw two functions simultaneously, simply connect them with
a logical OR. An example is
  x*y=1 | y=x.
Replacement of OR (|) with AND (&) results in an equation system.
It is also possible to draw inequalities, for example
  abs(x)+abs(y)<1.

Grapher for solutions of differential equations
Write y1 instead of y' and y2 instead of y'' and so on.

Grapher for systems of two differential equations
We want to study the damped oscillator wich is defined by the
equation y''+0.2y'+sin(y)=0. Let v(x):=y'(x). Put in
  y'(x)=v,
  v'(x)=-0.2v-sin(y).
Now choose v(y) (phase space) instead of y(x). Put in
  x0=0,
  y(x0)=-10:10,
  v(x0)=0.

Grapher for sequences
Recursion is possible. Take the recursive definition
  f(n) = if n=0 then 1 else 2f(n-1)
as an example. This has to be written in the form
  f(n) = if(n=0,1,2f(n-1)).
In the calculator line it is written in the form
  p:=fix({f,n|if(n=0,1,2f(n-1))}).
The function fix will memoize the values. Define
  fib:=fix({f,n|if(n=1|n=2,1,f(n-1)+f(n-2))}).
Now fib(40) will be calculated efficiently. Otherwise
the number of calculations would be in O(2^n).

The function sum(f,a,b) can be used to define a series.
The input for sum(from k=1 to n) 1/k is
  f(n) = sum({k|1/k},1,n)
or also
  f(n) = 1/n,
  g(n) = sum(f,1,n).

Grapher for complex valued functions
Let us take the function f(x) = exp(i*x) as an example.
The real part is expressed with
  f(x) = exp(i*x) or also with f(x) = re(exp(i*x)).
Analogously the the imaginary part is
  g(x) = im(exp(i*x))
and the absolute value is
  h(x) = abs(exp(i*x)).

Grapher for fractals
It is also possible to draw Julia sets.
For example put in f(z)=z^2-1 and z0=c.
r: out of bound radius
n: maximum number of iterations

Examples

Tangent line:
g(x) = diff(f,a)(x-a)+f(a)

Normal line:
h(x) = -1/diff(f,a)(x-a)+f(a)

Restriction of f to the interval (0,1):
g(x) = if(0<x&x<1,f(x),nan)

Function f on support (0,1):
g(x) = if(0<x&x<1,f(x),0)
g(x) = f(x)(x>0)(x<1)

Partial sum of the power series of the exponential function:
f(x) = sum({k|x^k/fac(k)},0,20)

Gamma function via parameter integral:
f(x) = int({t|t^(x-1)*exp(-t)},0,10)

Bifurcation diagramm for the logistic map:
f(x) = {r|pow({x|r*x*(1-x)},40,rand(0,1))}(x)

Mandelbrot set:
f(z) = {c|pow({z|z^2+c},10,0)}(z)

Newton fractal of x^3-1=0:
f(z) = pow({x|x-(x^3-1)/(3x^2)},10,z)
Click with the mouse in one of the basins to get the
corresponding root.

Table of prime numbers:
filter(isprime,1:100)

Count the number of primes:
count(isprime,1:100)

Twin primes:
filter({k|isprime(k)&(isprime(k+2)|isprime(k-2))},1:100)

Prime factorization of the numbers from 1 to 100:
table(factor,1:100)

Find cyclic multiplicative groups of integers:
filter({n|phi(n)=lambda(n)},1:100)

Continued fraction representation of the exponential function:
f(x) = 1+x/(1-reduce({t,k|x/k/(1+x/k-t)},rev(2:10)))

Interpolation polynomial of tanh:
f(x) = p(x); p:=ipp(table(tanh,-6:6))

Advanced

Defining functions
The calculator line is not just of use for calculating values,
but also for defining an arbitrary number of functions and
constants. For example, write the input
  p:={x|2*x}
and press the calc-button.
Now p(x) is available everywhere.

To define the function permanently, add the line
  evals("p:={x|2*x}");
to the end of the file "plot.js". If the function has to be
complex, add the line
  evalsc("p:={x|2*x}");
to the end of the file "cplot.js" instead.

A function may be programmed in JavaScript to extend the list
of predefined functions. To achieve the function is available,
add the following code to the end of the file "plot.js".

function p(x){
  return 2*x;
}
ftab.p=p;

Functions with two arguments
If you want to have the function p(x,y)=x*y, define p:={x,y|x*y}
in the calculator line and press the calc-button.

Alternatively define a function that takes a list instead.
So define in the calculator line p:={a|get(a,1)*get(a,2)}.
Therefore the function application is written p([x,y])
and not p(x,y).

Domain of a function
To draw the function f(x)=sin(x) only in the interval (0,2pi), write
  f(x) = if(0<x&x<2pi,sin(x),nan).
The expression stands for
  if 0<x and x<2pi, then sin(x), else not a number.

Solving equations
The equation f(x)=0 with f(x)=e^x+x^3-2x has to be solved. A plot
shows a zero of f in the interval [-2,-1] and that f is injective
in this interval. Therefore it is possible to draw the inverse
function with g(x)=inv(f,x,-2,-1). Now the zero is simply g(0).
To calculate the value, type g(0) in the calculator line.

Tables
To tabulate the function f(x)=x^2 on the intervall [0,4], write
  table({x|x^2},0:4:0.1)
in the calculator line. The input is equivalent to
  map({x|[x,x^2]},0:4:0.1).

Random numbers
Random numbers following a particular distribution are generated by
the inversion method. Firstly graph the function f(x) = inv(norm,x).
Now calculate f(rand(0,1)) in the calculator line. The results
are normal distributed random numbers. Now define
  a:=map({k|f(rand(0,1))},1:1000).
The function g(x) = cdf(a,x) will show the cumulative distribution
function for the random numbers in the list a.

Colors
With color(r,g,b) a function can be drawn in another color.
Each of the values r,g,b goes from 0 to 255.
If a number is too big oder negative, it is restricted
to the intervall from 0 to 255.
For example, you can write
  f(x) = sin(x), color(220,180,0).
Then the function will be drawn in yellow.
If bcolor(r,g,b) is calculated in the calculator line,
the background color of the canvas screen will be changed.

Number of pixels
If setw(1080) is calculated in the calculator line,
the width of the canvas screen will be increased to 1080 pixels.
The statement setw(w,h) will change width and height independently.

Coordinate system
The input of sp(x,y) (set position) in the calculator line
changes the origin of the coordinate system to the
pixel position (x,y).
In case there is a selection, "manual" has to be chosen.

Mouse
A click with the mouse on the canvas screen saves the
coordinates pointed to in the variables mx and my.
Thereafter it is possible to evaluate f(mx) in the calculator line.

Vertical lines
If the input is
  f(x) = 1/(x-2)/(x-4); vline(2,4),
vertical lines at the points x=2 and x=4
will be drawn in addition to the graph.

Discontinuities
The example
  f(x) = sgn(x); scatter([0,0]); box([0,1],[0,-1])
shows how to picture singular points and the absence of points.
A more complicated example:
  f(x) = floor(x);
    scatter(map({x|[x,x]},-6:6)),
    box(map({x|[x+1,x]},-6:6))

Piecewise linear interpolation
Example:
  f(x) = p(x); p:=fn([[0,0],[1,1],[2,0],[3,-1],[4,0]])
  
Piecewise linear interpolation, equidistant nodes
The last example has equidistant nodes. So one can use
a simpler algorithm. The usage is:
  fn(x0,d,[y0,y1,y2,y3,...,yn])
That means, we have points:
  [x0,y0], [x0+d,y1], [x0+2d,y2], [x0+3d,y3], ..., [x0+n*d,yn]
The last example simplifies to:
  f(x) = p(x); p:=fn(0,1,[0,1,0,-1,0])

Linear regression
Let a be a list of points [x,y].
Now t:=rl(a) calculates a linear regression and we have:
  y = t.ax*x+t.bx = t.ax*(x-t.mx)+t.my
  x = t.ay*y+t.bx = t.ay*(y-t.my)+t.mx
  t.mx: x mean
  t.my: y mean
  t.c: [t.mx,t.my] (center)
  r: sample correlation coefficient

Example:
  f(x) = p(x);
    a:=[[0,0],[1,2],[2,1],[3,4],[4,3],[5,4]],
    p:=fn(a), scatter(a), t:=rl(a), box(t.c)
  g(x) = t.ax*x+t.bx
  h(x) = (x-t.by)/t.ay

Saving images
The instruction save() generates an image of the canvas screen,
that can be saved. In some browsers the canvas screen
itself can be saved as an image. In this case, it is not necessary
to use save().

Animations
We want the amplitude of the sine function to swing between
-1 and 1. Define f(x) = r:=sin(a/10), r*sin(x). Now write ani() in
the calculator line and press the calc-button.

Closures
If you write {x;T(x)} instead of {x|T(x)}, a closure is created,
provied the anonymous function is contained in the term of another
anonymous function.

Let the continued fraction representation of the exponential
function serve as an example. Define
  p:=compose(map({k|{t;x/k/(1+x/k-t)}},2:10))
in the calculator line and press the calc-button.
The approximation of the exponential function is
  f(x) = 1+x/(1-p(0)).

Function grapher manual

Table of contents

Constants

Functions

Complex functions

Operators

Syntax and semantics

The graphers in detail

Examples

Advanced