About Functions and Graphs | Coordinate Systems and Descartes

What is a function?

A function is a rule that gives exactly one output for every input. The rule f(x) = x² turns 3 into 9 and −3 into 9 as well. The inputs a function accepts form its domain: 1/x has no value at 0, and ln x only works for positive numbers. Functions describe how one quantity depends on another, such as distance on time or the price on the amount you buy.

The Cartesian coordinate system

Two number lines cross at right angles at the origin (0, 0): the horizontal x-axis and the vertical y-axis. Every point in the plane gets an address (x, y): how far to go to the right, then how far up. The graph of a function is the set of all points (x, f(x)), so each x on the axis is lifted to the height of its value. The axes split the plane into four quadrants, numbered anticlockwise from the top right.

René Descartes and analytic geometry

In 1637 the French philosopher René Descartes published La Géométrie as an appendix to his Discourse on the Method. He showed that a curve can be written as an equation and an equation drawn as a curve, so problems of geometry could be solved with algebra. Pierre de Fermat had the same idea at about the same time. The name Cartesian comes from Descartes' Latin name, Cartesius, and the grid with two perpendicular axes was later shaped into the form we use today.

Other coordinate systems

A grid is not the only way to describe a point. Polar coordinates use a distance from the centre and an angle, which makes circles and spirals simple: r = 1 is a circle. Latitude and longitude locate places on the round Earth, and three axes x, y and z describe space. Logarithmic axes squeeze huge ranges into one view, so that exponential growth shows up as a straight line.

Frequently asked questions

What is an asymptote?

A line that a graph gets closer and closer to without reaching it. 1/x has a vertical asymptote at x = 0 and a horizontal one at y = 0, and tan x has a vertical asymptote at every odd multiple of π/2.

What does the parameter a do?

It turns one formula into a family of graphs. With a*x^2 the parabola gets narrower as a grows and flips over when a is negative; with sin(a*x) the wave gets faster. Moving the slider shows how each part of a formula shapes the curve.

Why is it called a Cartesian grid?

It is named after René Descartes, who wrote in Latin as Cartesius. His 1637 work showed how to link points to numbers, the idea behind every graph on this page.