1.4.8: Graphical conversions: summing up
This section started by looking at conversion graphs which were straight lines passing through the origin of the graph. The intercept in those cases was zero, and only one number – the gradient – was needed to describe the relationship between the quantities plotted on the horizontal and vertical axes. In the more general case, the graph is still a straight line with a constant gradient, but the line no longer goes through the origin. An extra number – the intercept – is used to pin the graph down to a particular location. You can think of a straight-line graph with a fixed slope being able to move vertically up or down, as in Figure 18. You can see that moving the graph vertically upwards increases the intercept, while moving it in a downwards direction decreases the intercept. If the straight line crosses the vertical axis below zero, the intercept is a negative number.
The general straight-line graph is described by the following formula:
value on the vertical axis = gradient x value on the horizontal axis + intercept
In the examples of conversion between different units you have seen this relationship in two forms:
where one quantity is directly proportional to another. The intercept is zero and the graph passes through the point (0,0);
where the quantities are related by a straight line, but zero in one system of units does not coincide with zero in another. The graph does not pass through the point (0,0). The formula includes a second fixed number-the intercept-which is the value at which the straight line meets the vertical axis.
Activity 9: Straight-line models
Spend a few minutes to note down some examples of relationships you think can be represented by straight-line graphs. Think about conversions you might make between different sorts of measurements or quantities, think about how bills are worked out. Which are directly proportional relationships and which are not?
Here are some ideas. Electricity, gas and telephone bills in the UK normally include a charge per unit plus standing charges which you must pay even if you have used nothing else. These give straight-line relationships (assuming there are no special deals or charge bands) between use and charge which lead to graphs with intercepts and constant gradients.
In contrast, there is a directly proportional relationship between costs and quantity for any item bought on a cost-per-unit basis.
The relationship between the angle of the hour hand (measured from the top of a clock) and the time is a directly proportional relationship, over a 12-hour period. Similarly for the minute hand over 1 hour.
The quantity of wallpaper or paint needed to decorate a room is directly proportional to the area of the walls.
The distance travelled is directly proportional to travel time if you travel at a steady speed. It is also roughly proportional to the amount of petrol used. Conversely, travel time is directly proportional to distance if you travel at a steady speed.