# Coordinates. Graphical representation of functions

*Coordinates. Coordinate system. Cartesian coordinates.*

Axes of coordinates. Origin of coordinates. Abscissa and

ordinate. Graphical representation of functions. Graph

of a functional dependence.

Axes of coordinates. Origin of coordinates. Abscissa and

ordinate. Graphical representation of functions. Graph

of a functional dependence.

** 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

coordinates

*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.*