ICSE Class 8 Maths Chapter 22 Linear Graphs

Chapter 22: Linear Graphs

Introduction

A graph consists of two mutually perpendicular number lines intersecting each other at zero. See the adjoining figure.

The given figure shows two number lines XOX' and YOY' such that:

(i) the lines are perpendicular to each other.

(ii) the lines intersect each other at zero (0).

x-axis: The horizontal number line XOX' is called x-axis.

y-axis: The vertical number line YOY' is called y-axis.

Co-ordinate axes: Taking together, the number lines XOX' and YOY' are called co-ordinate axes.

Axes is plural of axis.

Origin: The point at which the two axes intersect, is called origin and is denoted by letter 'O'.

Co-ordinate plane: The plane which contains both the co-ordinate axes, is called co-ordinate plane.

Teacher's Note

The coordinate system helps us locate any position on a flat surface, much like how we use street addresses and GPS coordinates to find locations in real life.

Co-Ordinates Of A Point

Consider a point P in the co-ordinate plane and PM perpendicular to x-axis.

1. The distance of point P, taken along x-axis and starting from origin O, is called the x-co-ordinate or abscissa of the point P.

i.e. OM = abscissa (or x-co-ordinate) of point P = x(let)

2. The distance of point P, taken along y-axis and starting from origin O, is called the y-co-ordinate or ordinate of the point.

i.e. PM = ordinate (or y-co-ordinate) of point P = y(let)

3. The abscissa (x-co-ordinate) and ordinate (y-co-ordinate) of a point together are called the co-ordinates of the point.

Thus, co-ordinates of a point = (its abscissa, its ordinate).

i.e. co-ordinates of the point P = (x, y)

In stating the co-ordinates of a point, the abscissa precedes the ordinate and both are enclosed in a bracket after being separated by a comma.

e.g., if the abscissa of point is 2 and its ordinate is -3, then its co-ordinates = (2, -3). Conversely, if the co-ordinates of a point are (a, b), then its abscissa = a and its ordinate = b.

4. For every point on x-axis, the value of its ordinate (i.e. y-co-ordinate) is zero and so the co-ordinates of every point on x-axis are of the form (x, 0).

Thus, the points (3, 0), (-4, 0), (0, 0), etc. lie on x-axis.

5. For every point on y-axis, the value of its abscissa (i.e. x-co-ordinate) is zero and so the co-ordinates of every point on y-axis are of the form (0, y).

Thus, the points (0, 3), (0, -5), (0, 0), etc. lie on y-axis.

6. Co-ordinates of origin are (0, 0) i.e. origin = (0, 0).

Teacher's Note

Understanding coordinates is like reading a map with grid squares - each location has a unique address given by its position from the horizontal and vertical reference lines.

Sign Convention

1. The abscissa (i.e. x-co-ordinate) of a point is positive, if the point is on the right of origin and is negative, if the point is on the left of origin.

Thus, each of the points (5, 7), (3, -6) and (6, 0) lies on the right side of origin as abscissae of these points are positive. And, each of the points (-5, 7), (-3, -6), (-3, 2) and (-6, 0) lies on the left side of origin as abscissae of these points are negative.

2. The ordinate (i.e. y-co-ordinate) of a point is positive, if the point is above the origin and is negative, if the point is below the origin.

Thus, each of the points (5, 7), (-3, 2) and (0, 5) lies above the origin as the ordinates of these points are positive. And, each of the points (5, -3), (-3, -6), (3, -6) and (0, -5) lies below the origin as ordinates of these points are negative.

Teacher's Note

Sign conventions in coordinates mirror how we describe directions in everyday life - right and up are positive, left and down are negative, just like north-south and east-west directions.

Quadrants

The two co-ordinate axes (x-axis and y-axis) divide the co-ordinate plane into four parts which are called quadrants.

As shown in the adjoining figure:

(i) In first quadrant, XOY, the abscissa and the ordinate both are positive.

(ii) In the second quadrant, X'OY, the abscissa is negative and the ordinate is positive.

(iii) In the third quadrant X'OY', the abscissa and the ordinate both are negative.

(iv) In the fourth quadrant, XOY', the abscissa is positive and the ordinate is negative.

Teacher's Note

The four quadrants help us organize all points on a plane, similar to how map makers divide the world into hemispheres to organize geographic information.

Test Yourself

1. Co-ordinate axes are two mutually ________________________ number ______________; which intersect each other at their ______________

2. XOX' is _______________________ number line and is called ______________

3. YOY' is _______________________ number line and is called ______________

4. The point of intersection of two number lines (axes) is called ______________

5. State, true or false:

(i) The point (5, 0) lies on x-axis ______________

(ii) The point (0, -8) lies on y-axis ______________

(iii) The point (0, 6) lies on x-axis ______________

(iv) The point (0, 0) lies on y-axis ______________

(v) If the abscissa of a point is zero, the point lies on x-axis ______________

(vi) If the ordinate of a point is zero, the point lies on x-axis ______________

6. Out of the points: (3, 5), (-3, 5), (3, -5), (-3, -5), (7, 8), (5, -4), (-6, 2), (8, 3), (5, -5), (-4, -4), (5, -3), (-6, -5), (-2, 3) and (6, 4) lie in:

(i) first quadrant: _____________, ________________________, ________________ and ______________

(ii) second quadrant: _____________, ________________________ and ______________

(iii) third quadrant: _____________, ________________________ and ______________

(iv) fourth quadrant: _____________, ________________________, ________________ and ______________

Plotting The Points

Example 1

Plot the points A (2, 3), B (-3, 2), C (-2, -2), D (1, -3), E (3, 0) and F (0, -1) on a graph paper.

Solution

Take a graph paper and on it, draw the co-ordinate axes XOX' and YOY' intersecting at origin O (as shown in the diagram). With a proper scale, mark the numbers on the two co-ordinate axes.

(i) For plotting A (2, 3); starting from origin O, move 2 units (abscissa) along x-axis, on the right of O, and then from there move 3 units (ordinate) along y-axis, above O. Mark the resulting point as A and write its co-ordinates (2, 3) near it.

(ii) For plotting B (-3, 2); starting from origin O, move 3 units along x-axis, on the left of O, and then from there, 2 units along y-axis, above O. Mark the resulting point as B and write its co-ordinates near it.

Similarly, mark the other points as shown.

Teacher's Note

Plotting points is like following a treasure map - first move the right distance horizontally, then move vertically to find the exact location of your treasure.

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