Wolfram|Alpha introduces a fundamentally new way to get knowledge and answers—
not by searching the web, but by doing dynamic computations based on a vast collection
of built-in data, algorithms, and methods.
PART 2 : Lets look at the mathematical searches and plotting
The following is not intended to be a handbook,
but only a listing of search examples
Plotting & Graphics
plot a function of one variable
plot x^3 - 6x^2 + 4x + 12
graph sin t + cos (sqrt(3)t)
plot 4/(9*x^(1/4))
specify an explicit range for the variable
plot e^x from x=0 to 10
plot a function where it is real-valued
real plot -(sqrt(25-y^2))
plot a special function
plot Ai(x)
plot multiple functions
plot sin x, cos x, tan x
plot a function on a logarithmic scale
log plot e^x-x
plot on a log-linear scale
log-linear plot x^2 log x, x=1 to 10
3D Plots
plot a function of two variables
plot sin x cos y
specify explicit ranges for the variables
plot x^2 y^3, x=-1..1, y=0..3
Equations
plot the solution to an equation in two variables
plot 3x^2-2xy+y^2=1
Inequalities
plot the region satisfying an inequality in two variables
plot |x|^3+|y|^3<1
plot a region satisfying multiple inequalities
plot x^2+y^2<1 and y>x
Polar Plots
draw a polar plot
polar plot r=1+cos theta
specify a range for the variable theta
polar plot r=theta, theta=0 to 8 pi
Parametric Plots
draw a parametric plot
parametric plot (cos^3 t, sin^3 t)
specify a range for the parameter
parametric plot (sin 10t, sin 8t), t=0..2pi
Number Lines
visualize a set of real numbers on a number line
number line 2, 3, 5, 7, 11, 13
show multiple sets on a number line
number line x^2>5, 0<=x<3
Algebra
Equation Solving
solve a polynomial equation
x^3 - 4x^2 + 6x - 24 = 0
solve a system of linear equations
x+y=10, x-y=4
solve an equation with parameters
solve a x^2 + b x + c = 0 for x
Polynomials
compute properties of a polynomial in several variables
x^3 + x^2 y + x y^2 + y^3
factor a polynomial
factor 2x^5 - 19x^4 + 58x^3 - 67x^2 + 56x - 48
Rational Functions
compute properties of a rational function
(x^2-1)/(x^2+1)
compute a partial fraction decomposition
partial fractions (x^2-4)/(x^4-x)
Simplification
simplify an expression
simplify x^5-20x^4+163x^3-676x^2+1424x-1209
simplify cos(arcsin(x)/2)
Vectors
compute properties of a vector
vector {2, -5, 4}
compute a cross product
{1/4, -1/2, 1} cross {1/3, 1, -2/3}
Matrices & Linear Algebra
do basic arithmetic on matrices
{{0,-1},{1,0}}.{{1,2},{3,4}}+{{2,-1},{-1,2}}
compute eigenvalues and eigenvectors of a matrix
eigenvalues {{4,1},{2,-1}}
Quaternions
compute quaternions
quaternion: 0+2i-j-3k
do calculations with quaternions
quaternion -Sin[Pi]+3i+4j+3k multiplied by -1j+3.9i+4-+3k
Finite Groups
get information about a finite group
alternating group A_5
ask about a property of a group
order of the monster group
do algebra with permutations
perm (1 2 3 4)^3(1 2 3)^-1
Finite Fields
compute properties of a finite field
Z/7Z
compute a specific property
number of primitive polynomials of GF(3125)
Domain & Range
compute the domain of a function
domain of f(x) = x/(x^2-1)
compute the range of a function
range of e^(-x^2)