You can have quadratic, or cubic, or even higher power
direct relationships. We can go back to the car acceleration example, where
the time taken to reach a certain speed was proportional to the *square of
the car’s speed*:

_{}

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In mathematical terms, the proportionality constant would be 0.0004388. We could write a general proportionality statement:

_{}

One of the neat things you can do with higher order
relationships is plot the squared or cubed values instead of just the value
itself. For instance, in the car speed graph above, we have plotted time, t
versus speed, s. We know that the relationship is that t is proportional to
the *square* of the speed. So one thing we can do is plot time versus the
*square of the speed*. We can do this by starting with our original data
for the graph:

Time, t, (s) |
0 |
1.7 |
2.9 |
4.4 |
6.4 |
8.6 |

Speed, s, (km/hr) |
0 |
60 |
80 |
100 |
120 |
140 |

Usually I like to have the first row of the table represent the data that goes on the x-axis, and the second row to represent the y-axis data, so let’s swap them around:

Speed, s, (km/hr) |
0 |
60 |
80 |
100 |
120 |
140 |

Time, t, (s) |
0 |
1.7 |
2.9 |
4.4 |
6.4 |
8.6 |

We want to plot time versus the square of the speed, so let’s redo the first row, by taking the square of every value. We should also relabel that row as well:

Speed |
0 |
3600 |
6400 |
10000 |
14400 |
19600 |

Time |
0 |
1.7 |
2.9 |
4.4 |
6.4 |
8.6 |

When we plot this graph, we get a very interesting result:

It’s basically a *straight line!* This is a
characteristic of proportional relationships. If you have a relationship where
_{}, then
plotting A versus B will give you a straight line. If you have a relationship where
_{}, then
plotting A versus B^{2} will give you a straight line. If you have a
relationship where _{} then if you plot A versus B^{3}
you’ll get a straight line.

This also works for fractional or negative powers. Say
we’ve got a relationship where _{}. This is the same as saying_{}. If we plot a
graph of A versus _{} then you’ll get a straight line.

To explain why this happens let’s consider one case – a direct cubic relationship between x and y:

_{}

Because this relationship is proportional, we can write an equation relating ‘x’ and ‘y’ to each other using a constant of proportionality, k:

_{}

If we just plotted y versus x then we’d get a graph
with a curvy line in it. Remember that you get a straight line from any
equation where one thing to the power 1 equals some constant times another
thing to the power 1. The key – the powers of both variables must be the same
– ‘1’. If we *replace* the _{} with something called ‘z’ say, we
get a new equation:

_{}

This is a *linear equation*, which will give a
straight line graph. So by plotting ‘y’ values against ‘z’ values, which
represent x^{3}, we get a straight line.