ICSE Class 7 Maths Chapter 20 Graphs Plotting of points Co Ordinates

Chapter 20

Graphs: Plotting Of Points (Co-Ordinates)

Elementary Treatment

In the given figure, two number lines XOX' and YOY' are drawn perpendicular to each other, both intersecting at their zeroes, marked as O.

The horizontal number line XOX' is called the X-axis.

The vertical number line YOY' is called the Y-axis and the point O, where the two axes meet (zero of both the axes), is called the origin.

Together both, the X-axis and the Y-axis, are called the co-ordinate axes and, also, rectangular co-ordinates. [Axes is the plural of axis]

The part of the X-axis OX, i.e., on the right of origin O, is called the positive X-axis and its part on the left of O, i.e., OX', is called the negative X-axis.

The part above O(OY) on the Y-axis is called the positive Y-axis and its part below O(OY') is the negative Y-axis.

Teacher's Note

Understanding coordinate axes is like using a street address system - just as "5th Avenue and 42nd Street" pinpoints a location in a city, coordinates pinpoint exact locations on a plane.

Representation Of A Point On A Plane

Taking the co-ordinate axes as the reference lines, the distance of a point on the X-axis, starting from origin O, is called the x-co-ordinate of the point.

The distance of the point from O on the Y-axis is called its y-co-ordinate.

The x and y-co-ordinates together are called the co-ordinates of a point.

In order to write the co-ordinates of a point, its x-co-ordinate is written first, then y-co-ordinate is written and then they are enclosed in a bracket.

i.e., the co-ordinates of a point = (its x-co-ordinate, its y-co-ordinate).

For example:

If x-co-ordinate of a point is 4 and its y-co-ordinate is 7, then the co-ordinates of the given point are (4, 7).

The co-ordinates of a point are also termed as an ordered pair.

Reason:

As discussed above, (4, 7) are the co-ordinates of a particular point. Do (7, 4) also represent the co-ordinates of the same point? No.

In (4, 7), the x-co-ordinate is 4 and the y-co-ordinate is 7; whereas in (7, 4), the x-co-ordinate is 7 and the y-co-ordinate is 4.

∴ (4, 7) and (7, 4) do not represent the same point, infact these two ordered pairs represent two different points in a co-ordinate plane.

Thus, in an ordered pair the change in order of its constituent elements (numbers) changes the ordered pair.

Teacher's Note

The order of coordinates matters just like "10 Main Street, Apt 5" is different from "5 Main Street, Apt 10" - swapping the numbers creates a completely different location.

In the adjoining figure, for point A, its horizontal distance from the origin O is 3, therefore, its x-co-ordinate is 3.

The vertical distance of the point A from the origin O is 2, therefore, its y-co-ordinate is 2.

Hence, the co-ordinates of the point A = (3, 2).

Similarly, the co-ordinates of point B = (1, 3) and the co-ordinates of point C are (-4, -3).

Marking (Plotting) Of The Points

For plotting the points, we use squared paper, called graph paper.

Two mutually perpendicular number lines (one horizontal and another vertical) are drawn. The point of intersection of these lines is marked as O (the origin) which is 0 (zero) of both the number lines. The horizontal line (the x-axis) is marked as XOX' and the vertical line (the y-axis) is marked as YOY'.

The following points are plotted on the graph paper, drawn above, on which the axes are drawn with the negative and the positive numbers marked.

A(3, 4), B(5, 2), C(-1, 4), D(-2, -3), E(4, 0) and F(0, -6).

For plotting the point (3, 4), start from origin O and measure 3 units to its right on the x-axis for +3 and from there 4 units upwards for +4.

We reach at point A = (3, 4).

[The co-ordinates of the point are written near its position. The marking of the point is usually done by a cross, i.e., x or by a dark point or by a circle.]

For B(5, 2), we start from O towards right and measure 5 units for +5. From that position, we move 2 units upwards for +2. We reach at B which is the position of the point whose co-ordinates are (5, 2).

In the same way, other points C, D, E and F are marked on the graph paper.

Teacher's Note

Plotting points is like following GPS directions - you move horizontally first (like moving east), then vertically (like moving north) to reach your exact destination on a map.

Abscissa And Ordinate

The x-co-ordinate of a point is called its abscissa and the y-co-ordinate of the point is called its ordinate.

If the co-ordinates of a point are (4, 3), its abscissa = 4 and its ordinate = 3.

Conversely, if the abscissa of a point is -2 and its ordinate is 5, then the co-ordinates of the point are (-2, 5).

The co-ordinates of the origin O are (0, 0).

Quadrants

The co-ordinate axes, x-axis, and y-axis, divide a plane (graph) into four equal parts. Each of these four parts is called a quadrant.

As is clear from the figure on the previous page, the first quadrant is the region XOY, the second quadrant is the region X'OY, the third quadrant is the region X'OY' and the fourth quadrant is the region XOY'.

1. For a point in the first quadrant, its abscissa and ordinate both are positive. Point (2, 3) belongs to the first quadrant, since its abscissa (2), and ordinate (3) both are positive.

2. For a point in the second quadrant, its abscissa is negative and ordinate is positive. Point (-2, 1) belongs to the second quadrant, since its abscissa (-2) is negative and ordinate (1) is positive.

3. For a point in the third quadrant, its abscissa and ordinate both are negative. Point (-3, -3) belongs to the 3rd quadrant.

4. For a point in the fourth quadrant, its abscissa is positive and ordinate is negative. Point (3, -2) belongs to the 4th quadrant (why ?).

1. For a point on x-axis, its ordinate (y-co-ordinate) is always zero and so the co-ordinates of a point on x-axis are of the form (x, 0). Points (3, 0), (5, 0), (0, 0), (-7, 0), (-11, 0), etc., all lie on x-axis, because the ordinates of all these points are zero.

2. For a point on y-axis, its abscissa (x-co-ordinate) is always zero and so the co-ordinates of a point on y-axis are of the form (0, y). Points (0, 4) (0, 6) (0, 0), (0, -5), (0, -10), etc., all lie on y-axis, because the abscissae of all these points are zero.

Teacher's Note

The quadrant system is like dividing a city into four zones - northeast, northwest, southeast, and southwest - each with distinct characteristics based on which direction you move from the center point.

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