Question 3 on Paper 2 deals with both of these topics, which are connected in that they both deal with the Cartesian plane. The Cartesian, or coordinated, plane was first introduced by the French mathematician, Decartes, early in the 17th century. This was a major leap forward, as it allowed, for the first time, the power of algebra to be used for dealing with problems in geometry, and, conversely, a geometric interpretation to be put on many algebraic problems. It was this latter use that fascinated Descartes, but our concern nowadays is more with the former.
By the time students start studying Leaving Cert maths, they have normally been dealing with co-ordinate geometry for two to three years. This familiarity leaves most well disposed towards The Line. On this course, the basic formulae and methods from Junior Certificate, e.g., distance, midpoint, slope, equation of a line, area of a triangle, are augmented by a few new formulae and methods. Whereas these command most attention, the more basic ideas should not be neglected.
Transformations, or linear transformations to give them their full title, are functions which map points in the plane to other points in the plane. However, they do so in a very regular and organised way, so that certain properties are always preserved. For example, lines are always mapped to lines, and parallel lines are always mapped to parallel lines. But other properties, e.g., distance, are not necessarily preserved. On our course, we investigate, for specific transformations, such features of points, lines and shapes in the plane under the action of a transformation.
Topic Structure: The Line
The study of The Line for Leaving Cert can be broken down into the following sections:
1. Basic Co-ordinate Geometry Results
2. Leaving Cert Formulae
3. Leaving Cert Methods
Topic Structure: Transformations
The study of Leaving Cert Transformations can be broken down into the following sections:
2. Proofs and Investigations
This 'Linear Transformations' section has a slightly different emphasis than what else appears in the study notes, but interested students will find it useful and perhaps gain additional insight into familiar material.
This excellent link is to an interactive exercise in the equations of lines, converting from one form to another, including parametric, and plotting and recognising lines.