## Lesson 12: Functions (I)

June 22, 2012

The French philosopher and mathematician, René Descartes (1596-1650) developed his philosophy on the idea of needing a starting point on which all other knowledge can be based: I think, therefore I am. In the world of mathematics he is the founder of analytic geometry, which uses a pair of perpendicular lines that cross at a point called the origin as its basis, and is also referred to as the Cartesian coordinates system.

First recall some concepts about Cartesian coordinates on the plane.

### 1. The Cartesian plane

We will begin with vocabulary.

First, a coordinate. A coordinate is a number  that labels a point on a line.

The coordinate 0 is called the origin of coordinates.

Points to the right of 0 are labelled with positive numbers: 1, 2, 3, etc.  Points to the left of the origin are labelled with negative numbers: −1, −2, −3, etc. Those coordinates are the “addresses” of those points.

A coordinate axis is a line with coordinates.

Now, to label the points in a plane, we will need more than one coordinate axis, and we place a second at right angles to the first.

Points above the origin have positive coordinates; points below have negative coordinates.

Those lines are called rectangular coordinate axes, because they are at right angles to one another; the coordinates on them are called rectangular coordinates.

Hence we have the name coordinate geometry or, as it is often called, analytic geometry.

The rectangular coordinates of a point are an ordered pair, (x, y).

The coordinates of the origin O are (0, 0). We don’t move right or left and we don’t move up or down. We will see that 0 is an extremely important coordinate.  It means that the point is on one of the axes.

The horizontal axis is called the X-axis or axis of abscissas, the vertical axis is called the Y-axis or axis of ordinates  and the fixed point O is called the origin.

This system of reference can be used to determine the position of any point on a set of axes by an ordered pair of numbers, which are usually written in brackets and separated by a comma.

The numbers in each ordered pair of numbers are called the coordinates of the corresponding point. The first number is called the x-coordinate (or abscissa) and the second the y-coordinate (or ordinate).

We always write (x, y); x is named abscissa and y ordinate

To make it easy to talk about where on the coordinate plane a point is, we divide the coordinate plane into four sections called quadrants, labelled counter-clockwise: The first, the second, the third, and the fourth

 In Roman numbers Arabic numbers

Points in Quadrant 1 have positive x and positive y coordinates.

Points in Quadrant 2 have negative x but positive y coordinates

Points in Quadrant 3 have negative x and negative y coordinates.

Points in Quadrant 4 have positive x but negative y coordinates.

In the following links you can practice several skills: