Cambridge International Mathematics

266 Coordinate geometry (Chapter 12)

3 Find the gradient of the line segment joining the following pairs of points:

a (2,3) and (7,4) b (5,7) and (1,6) c (1,¡2)and (3,6)

d (5,5) and(¡ 1 ,5) e (3,¡1) and(3,¡4) f (5,¡1)and (¡ 2 ,¡3)

g (¡ 5 ,2) and (2,0) h (0,¡1) and(¡ 2 ,¡3)

4 On the same set of axes draw lines through(1,2)with gradients of^34 ,^12 , 1 , 2 and 3.

5 On the same set of axes draw lines through(¡ 2 ,¡1)with gradients of 0 ,¡^12 ,¡ 1 and¡ 3.

USING GRADIENTS

In real life gradients occur in many situations, and can be interpreted in a variety of

ways.

For example, the sign alongside would indicate to motor vehicle drivers that there is an

uphill climb ahead.

Consider the situation in the graph alongside where a motor vehicle

travels at a constant speed for a distance of 600 km in 8 hours.

Clearly, the gradient of the line =

vertical step

horizontal step

=

600

8

=75

However, speed=

distance

time

=

600 km

8 hours

=75km/h.

So, in a graph of distance against time, thegradientcan be interpreted as thespeed.

In the following exercise we will consider a number of problems where gradient can be interpreted as a rate.

EXERCISE 12D.2

1 The graph alongside indicates the distances and corresponding

times as Tan walks a distance of 50 metres.

a Find the gradient of the line.

b Interpret the gradient found ina.

c Is the speed of the walker constant or variable? What evi-

dence do you have for your answer?

2 The graph alongside indicates the distances travelled by a

train. Determine:

a the average speed for the whole trip

b the average speed from

i AtoB ii BtoC

c the time interval over which the speed was greatest.

600

400

200

2468

time (hours)

distance (km)

O

20 40

50

25

distance (m)

time (s)

O

distance (km)

time (hours)

A

B

C

D

(1,¡65) (2,¡180)

(5,¡560)

(6,¡630)

O

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y:\HAESE\IGCSE01\IG01_12\266IGCSE01_12.CDR Friday, 3 October 2008 12:11:39 PM PETER