When we calculated the speed for this data, we saw that it was 10 m/s. The rate of change is also 10. Therefore, for a distance/time graph the rate of change is equal to the speed.


In addition, we can see that as the time spent travelling increases, the distance travelled will also increase.


This type of relationship is called a directly proportional relationship.


A directly proportional relationship exists between two variables when they both begin at 0 and then as one measurement increases the other one also increases.


The graph of a directly proportional relationship always goes through (0, 0).


This type of graph will always form a straight line because distance travelled is directly proportional to the time spent travelling. The more time you spend travelling the greater the distance you will travel.


Discuss and discover


Can you think of any other proportional relationships? (Hint: Numbers of hours worked is directly proportional to the money earned by a worker.)


Now consider measuring the distance of a second object over the same time. The results are shown in the table below.


Time (s) Distance from start (m)


0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 120 140 160


For every 1-second increase in time, there is a 20-metre increase in the distance travelled. This increase stays constant during the 8 seconds recorded. We can therefore say that the object is travelling at a constant speed of 20 m/s.


If we calculate the speed using the distance/time formula for this object: 80 ÷ 4 = 20 m / s We see that this object is moving faster than the fi rst one. It has covered the same distance in a shorter time. Let’s look at the graph for this object (shown in blue). Notice that the graph is still a straight line.


y


The rate of change is 20. For every 1 unit across, the graph goes up 20. The speed is 20 m/s. We can also see the line is much steeper than the red graph above. The steepness of the line is referred to as the slope of the line. The slope of the line is determined by the rate of change.


Slope = rate of change


The steeper the line, the greater the slope and hence the greater the rate of change. A horizontal line has a slope of zero and hence no rate of change.


80 70 60 50 40 30 20 10


0 1 2 3 4 5 6 7 Time (seconds) 8 x Distance travelled against time y


80 70 60 50 40 30 20 10


0 1 2 3 4 5 6 7 Time (seconds) 8 x Distance travelled against time


222


Linking Thinking 1


Distance (metres)


Distance (metres)


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