Lesson 12: Functions (I)June 22, 2012
A little of History
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
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: