Cambridge International AS and A Level Mathematics Pure Mathematics 1

The gradient of a line

P1^

2

Plotting, sketching and drawing

In two dimensions, the co-ordinates of points are often marked on paper and

joined up to form lines or curves. A number of words are used to describe

this process.

Plot (a line or curve) means mark the points and join them up as accurately as

you can. You would expect to do this on graph paper and be prepared to read

information from the graph.

Sketch means mark points in approximately the right positions and join them up

in the right general shape. You would not expect to use graph paper for a sketch

and would not read precise information from one. You would however mark on

the co-ordinates of important points, like intersections with the x and y axes and

points at which the curve changes direction.

Draw means that you are to use a level of accuracy appropriate to the

circumstances, and this could be anything between a rough sketch and a very

accurately plotted graph.

The gradient of a line

In everyday English, the word line is used to mean a straight line or a curve. In

mathematics, it is usually understood to mean a straight line. If you know the

co-ordinates of any two points on a line, then you can draw the line.

The slope of a line is measured by its gradient. It is often denoted by the letter m.

In figure 2.2, A and B are two points on the line. The gradient of the line AB is

given by the increase in the y co-ordinate from A to B divided by the increase in

the x co-ordinate from A to B.

y

x

(2, 4)

A

B(6, 7)

O

Figure 2.2

θ

Gradient m = 7462 −− = (^43)
7 − 4 = 3
θ (theta) is the Greek letter ‘th’.
α (alpha) and β (beta) are also
used for angles.
6 − 2 = 4