To plot a point in a plane, two position numbers are needed. These position numbers are known as coordinates. The horizontal position number is called the x-coordinate and the vertical position number is called the y-coordinate. The x-coordinate and the y-coordinate are also known as abscissa and ordinate.

To locate a point in a plane (two-dimensional space), a horizontal number line and a vertical number line are used.

The horizontal number line is called the X-axis. The vertical number line is called the Yaxis.

The intersection of two number lines, the point O, is called the origin.

The X-axis and Y-axis separate the plane called a coordinate plane into four regions. These regions are called quadrants.

To locate a point in the coordinate plane two position numbers, written in the form of an ordered pair, are used.

x = left hand side position number

y = right hand side position number

Writing on ordered pair, we enclose the two numbers within the round brackets. The two numbers are separated by a comma.

Ordered pair = (x, y)

Note: The origin has coordinates (0, 0), (2, 5) and (5, 2) are not the same.

On the Y-axis all x-coordinates are zero. Horizontally, to the right-hand side of the Y-axis

x > 0 or x is positive. Horizontally to the left hand side of the Y-axis is x < 0 or x is negative.

On the X-axis all y-coordinates are zero. Vertically above the X-axis, y > 0 or y is

positive. Vertically below the X-axis, y < 0 or y is negative.

The axes divided the plane into four quadrants, numbered as shown in Fig.

(1) Using a graph paper, plot the following points. (Scale : 2 cm = 1 unit each axis)

(a) A (2, 0) (b) B (– 3, 2) (c) C (– 2, 3) (d) D (0, 3) (e) E (0, – 3) (f) F (2, 1) (g) G (1, 2)

(2) Vertices of a triangle ABC are A(4, 0), B (2, 2) and C (0, 0). Draw and label the triangle ABC (Scale : 2 cm = 1 unit on each axis)

(3) Plot the points A (3, 5), B (2, 3), C (3, 1) and D (4, 3). Join AB, BC, CD and AD. Name the geometrical figure that you have drawn.

(4) Draw a figure whose angular points are (10, 12), (– 2, – 4) and (2, – 8) and find its area.

Show that the points (0, 2), (– 1, –1), (1, 5), (2, 8) and (– 3, –7) all lie on a straight line.

An expression which involves only a variable and constants is called a “function of that variable.” Thus if an expression involves only a variable x and constants, then the expression is called a “function of x.”

This is written f (x).

Thus if y = f (x) then y must be an expression which contains only x and constants. Hence, 

y = 3x + 4,

y = 5x2– 6x – 9,

y = a x3 + bx2 + cx + d, are expressions where y is a function of x.

Thus if in an expression y=f(x) a value is given to x, then there will be a corresponding value of y.

The values given to x and the corresponding derived values found for y can be plotted and a graph drawn. The resulting curve is called the graph of the function f(x). Very often this is called the graph of the equation y = f (x).

It will be found that functions can be classified into certain graph forms. Thus :

1. Any equation of the first degree (i.e., the terms including the variable are of the first degree) always gives a straight line graph.

2. Any equation of the second degree always gives a curve which is called a parabola. (This is the curve which any projectile traces out when fired through the air).

Slope-intercept Form

The y-coordinate of the point where a non-vertical line intersects the Y-axis is called the y intercept of the line. 

The equation of the line having slope m and y intercept c is

m = (y – c)/ (x – 0)

mx = y – c

y = mx + c

Equations of the Second Degree

We explained this equation in Sec.2.2. The following examples illustrate the techniques involved.