The most common graphical representation of a function involves the use of

rectangular Cartesian coordinates. These involve two mutually perpendicular

axes on each of which (unless otherwise stated) the same length scale is used

to represent real numbers. The horizontal axis is the x -axis, with positive x to

the right of the point of intersection of the two axes which is taken as the

origin, and negative x to the left. The vertical axis is the y -axis, with positive

y above the origin and negative y below it. A typical point P in the (x , y )-plane shown in Fig. 9 is identified by its

x -coordinate a and its y -coordinate b, with a the number of length units P is

distant from the y-axis, and b the number of length units P is distant from

the x-axis, with due regard to sign. Thus Q is the point (2, 1) and R is the

point (/3, /2). The number pair (a , b ) is called an ordered pair because the order in which a and b appear is important. Interchanging a and b in the

ordered pair (a , b ) to give (b, a ) changes the point represented by this

notation. The distance AB between points A (x1, y1) and B (x2, x2) in Fig. 9 is the

length of the straight line AB so, by Pythagoras’ theorem,


jx2x1j2 jy2y1j2

and hence

AB[(x2x1) 2(y2y1)

Thus, by way of example, the distance QR in Fig. 9 is



The graph of


is the straight line shown in Fig. 10. The number m is called the gradient

(slope) of the line and tanu/m , where u is the angle between the line and the x -axis, as measured in Fig. 10. The number c is the intercept of the line on

the y -axis. If m/0, then the line y/mx/c is a monotonic increasing function, and if mB/0, then it is a monotonic decreasing function. When m/0 the equation of the line reduces to the constant function y/c , whose graph is the dashed line in the figure parallel to the x -axis and passing through the

point c on the y-axis. Lines parallel to the y-axis are of the form x/c , where c is a constant.