2.2 Graphical representation of functions 33
For each value of xthere exists a value of y. A table can be drawn, such as Table 2.1,
giving values of ycorresponding to a set of values of x. In addition, each pair of
numbers (x, y) in the table may be regarded as defining the position of a point in a
plane, and can be plotted in a graph as in Figure 2.1.
Table 2.1
xy
− 312
− 25
− 10
0 − 3
1 − 4
2 − 3
30
45
512
The function given by equation (2.3) is called a quadraticfunction because the
highest power of xis the square (in plane geometry, quadrature is the act of squaring;
that is, finding a square whose area is equal to that of a given figure). It is an example
of a general class of functions called polynomials; polynomials and other algebraic
functions are discussed in the following sections.
0 Exercises 7, 8
The cartesian coordinate system
2The position of a point in a plane is specified uniquely by its coordinatesin a given
coordinate system. The most generally useful system is the cartesian (rectangular)
coordinate system shown in Figure 2.2.
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− 3 − 2 − 112345
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............................................................................................................................................................................................................................................................................................................................................................................................................Figure 2.1
2René Descartes (1596–1650), or Renatus Cartesius, French philosopher and mathematician. He attributed his
search for a universal mathematics to a mystical experience in 1619 in which ‘full of enthusiasm, I discovered the
foundation of a wonderful science’. He developed the relation between algebra and geometry in his Géométrie,
published as an appendix to the Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les
sciences(Discourse on the method of good reasoning and seeking truth in the sciences), 1637. Before Descartes,
the quantities x, x
2, and x
3were always associated with the geometric concepts of line, area, and volume. Descartes
discarded this restriction, ‘the root (x), the square, the cube, etc. are merely magnitudes in continuous proportion’.
His work marks the beginning of modern algebra. Descartes introduced the convention of using letters at the
beginning of the alphabet (a, b,=) for constants and parameters, and letters at the end (x, y, z) for unknowns
and variables. The Géométriecontains a formulation of the fundamental theorem of algebra, and the first use of
the words ‘real’ and ‘imaginary’ in the context of complex numbers.
Coordinate geometry was also developed by Fermat at about the same time as Descartes, but his work was
not published until 1679, after his death. Pierre de Fermat (1601–1665), lawyer at the provincial parliament of
Toulouse, made important contributions to the theory of numbers and to the calculus.