Coordinate System and Graphs
Coordinate
System
On a graph paper, draw two lines XOX’ and YOY’ mutually perpendicular to each other intersecting at O.
1. The horizontal line XOX’ is called X-axis.
2. The
vertical line YOY’ is called Y-axis.
3. XOX’ and
YOY’ together are called coordinate axes.
4. The point
of intersection of XOX’ and YOY’, i.e., O is called the origin.
OX and OY
are in the positive directions of X-axis and Y-axis respectively and OX’ and OY’
are in the negative directions of X-axis and Y-axis respectively.
Quadrants
The two axes
divide the plane into four parts each of which is called a quadrant. Quadrants
are named I, II, III and IV in anti-clockwise direction. This representation of
axis, quadrants, etc., is called coodinate plane or XY plane and the axis are
called coordinate axes.
The sign of
the coordinates are given below:
1. In first quadrant, x is positive, y
is positive.
2. In second quadrant, x is negative, y
is positive.
3. In third quadrant, x is negative, y
is negative.
4. In fourth quadrant, x is positive, y
is negative.
Coordinates of a Point
Let P be a
point on the coordinate plane. Draw AP perpendicular to X-axis and BP
perpendicular to Y-axis then
BP is called the x-coordinate or abscissa of point
P and is denoted by x.
AP is called the y-coordinate or ordinate of point P and
is denoted by y.
The abscissa and the ordinate of a point are called the
coordinates of point P and is represented as an ordered pair (x, y).
Coordinates
of the origin is (0, 0), coordinates of any point on X-axis is (x, 0) and on
Y-axis is (0, y). Thus, the coordinate of a point specify the position of a
point with reference to the coordinate axes.
Plotting of Points
For plotting
points on the graph paper, follow the given steps.
1. Draw XOX’
and YOY’.
2. Choose a
scale on both the axes and label it.
3. To plot a
point, B (4, –3) move 4 units along x-axis towards right of zero and then move
3 units along the direction of negative y-axis and mark the point B.
The coordinates
of point B are (4, –3).
Example
1: Plot the
following points on a graph paper and state the quadrant on which they lie.
A (5, 3), B
(–3, 2), C (7, 7), D (–4, –5), E (6, –2), F (0, 5), G (–6, 0)
Solution: On a graph paper, draw coordinates
XOX’ and YOY’ intersecting each other at O. The given points are plotted on the
graph paper as shown. The quadrants on which the given points lie are
A (5, 3) —
1st quadrant
B (–3, 2) —
2nd quadrant
C (7, 7) —
1st quadrant
D (–4, –5) —
3rd quadrant
E (6, –2) —
4th quadrant
F (0, 5) —
y-axis
G (–6, 0) —
x-axis
Graphs of Linear Equations
In order to
draw the graph of a linear equation, proceed as shown.
1. In the
given equation, give some value to one variable and find the corresponding
value of the other variable. Express these values in a tabular form.
2. Plot
these points and join any two points by a straight line. To determine a
straight line only two points are required but it is suggested that three coordinates
are determined.
Example 2: Draw the graph of each of the
following.
5x – 4y = 15
Solution:
The given equation
is 5x – 4y = 15, i.e., x = (15 + 4y)/5 .
The table
for the solution of the equation is given as:
x
|
3
|
7
|
-1
|
y
|
0
|
5
|
-5
|
Plot the
points A (3, 0), B (7, 5), C (–1, –5) to get the graph of the equation.
