Illustrating direct and inverse proportion
How do we know if a graph represents either direct or inverse proportionality? We look for certain features of the graph and perform some test calculations to help us decide.
Look at this graph:
The graph shows that the distance travelled and the time taken are proportional, but how do we know that?
Notice that the graph is a straight line starting from the origin. When both of these features are present we know that the two quantities on the graph must be directly proportional. However, take note of the scales on the graph, if they do not start at (0,0) or if they are not linear take care.
Look at the values on both of the axes - when the distance axis is 4 the time axis is 2, when the distance axis shows 8 the time axis shows 4. This means that when one of the variables doubles the other variable also doubles, this is the test for proportionality. If this condition is true and the graph is a straight line then we must have a directly proportional relationship.
The values on the axes also serve another important purpose as they allow us to discover the constant of proportionality from the graph so that we can describe the relationship using an equation. Look at the time value when the distance is 40, you should be able to see that it is 20. This means that the constant of proportionality which is linking distance to time is 40 ÷ 20 = 2.
\({distance}∝{time}\)
Removing the proportional sign and adding the constant of proportionality gives:
\({distance}={k}\times{time}\)
Rearranging we get:
\({k}=\frac{distance}{time}=\frac{40}{20}={2}\)
So finally we can write:
\({distance}={2}\times{time}\) or \({d}={2t}\)
If we want to make \({time}\) the subject of the equation, we would have to divide both sides of the equation by 2. This would give:
\({t}=\frac{d}{2}\)
Question
Which of the following graphs show direct proportion? For the ones that do, calculate the constant of proportionality.
Graph B shows a proportional relationship while Graph A does not. We know Graph A is not proportional because when the \({y}\)-axis shows, for example, 4 the \({x}\)-axis shows 0. If we double 4 to get 8 on the \({y}\) -axis, we would expect the \({x}\)-axis value to also double. As it does not, the graph does not show a proportional relationship.
We know that Graph B does show a proportional relationship because if you look at the value on the \({x}\)-axis when the \({y}\)-axis is 75, we see that it is 50. When we double the value on the \({y}\)-axis to get 150 and look for the corresponding value on the \({x}\)-axis, we see that is it 100. So it has also doubled and therefore the relationship is proportional.
To find the constant of proportionality, we need to divide a value on the \({y}\)-axis by a corresponding value on the \({x}\)-axis: 300 ÷ 200 = 1.5
We can therefore write:
\({income}={1.5}\times{books~sold}\)
Inverse proportion
To recognise inverse proportionality we need a different set of rules. Firstly we must note that straight-line graphs cannot show inverse relationships, so we are looking for curves.
This graph shows an inverse proportional relationship, but how do we know? The characteristic feature of inverse proportionality is that when one variable doubles the other one halves. Let’s see if this is true.
When the y-axis of the graph reads 1, the corresponding value on the x-axis is also 1. If we double the value on the x-axis we get 2, and this corresponds to 0.5 on the y-axis. This confirms that the graph shows inverse proportionality as when one variable doubles, the other one halves.
Written algebraically we can say:
\({p}∝\frac{1}{g}\)
Removing the proportional sign and adding a constant of proportionality we have:
\({p}=\frac{k}{g}\)
Rearranging for \({k}\):
\({k}={p}\times{g}\)
Using a set of corresponding values from the graph (we have chosen to use 2 and 0.5) gives:
\({k}={0.5}\times{2}={1}\)
So the constant of proportionality is 1
Finally we can say:
\({p}=\frac{1}{g}\)
