Coordinates. Graphical representation of functions

Coordinates. Two mutually perpendicular straight lines XX’ and YY’ ( Fig.1 ) form a coordinate system, called Cartesian coordinates. Straight lines XX’  and YY’ are called axes of coordinates. The axis  XX’ is called an x-axis, the axis YY’ – an y-axis. The point O of their intersection is called an origin of coordinates. An arbitrary scale is selected on each axis of coordinates.

Find projections P and Q of a point M to the coordinate axes  XX’  and  YY’ . The segment OP on the axis XX’ and a number x, measuring its length according to the selected scale, is called an abscissa or  x-coordinate of a point  M ; the segment OQ on axis YY’ and a number  y , measuring its length - an ordinate or  y-coordinate of a point M. Values x = OP and  y = OQ  are called Cartesian coordinates ( or simply – coordinates ) of a point M. They are considered as positive or negative according to the adopted positive and negative directions of coordinate axes. Usually positive abscissas are placed by right on an axis XX’ ; positive ordinates –  by upwards on an axis  YY’. On  Fig.1  we see: a point M  has an abscissa  x = 2, an ordinate y = 3; a point  K  has an abscissa  x = – 4 , an ordinate  y = – 2.5. This can be written as: M ( 2, 3 ),  K ( – 4, – 2.5 ).  So, for each point on a plane a pair of numbers (x, y) corresponds, and inversely, for each pair of real numbers (x, y) the one point on a plane corresponds .

Graphical representation of functions.

To represent a functional dependence  y = f ( x ) as a graph it is necessary:
 1)  to write a set of values of the function and its argument in a table:

 2) To transfer the coordinates of the function points from the table to a coordinate system,
      marking according to the selected scale a set of x-coordinates on x-axis and a set of
      y-coordinates on  y-axis ( Fig.2 ). As a result a set of points A, B, C, . . . , F will be
      plotted in our coordinate system.

 3)  Joining marked points A, B, C, . . . , F  by a smooth curve, we receive a graph of the given
      functional dependence.

Such graphical representation of a function permits to visualize a behavior of  the function, but has an insufficient attainable accuracy. It’s possible, that intermediate points, not plotted on a graph, lie far from the drawing smooth curve. Good results also depend essentially on a successful choice of scales. That is why, you should define a graph of a function as a locus, coordinates of points of which M (x, y)  are tied by the given functional dependence.