### Functions and graphs: Relations and functions

### Example 1 of a linear relation: Height prediction

One often works in science with mathematical models that describe experimental relationships between quantities and make it possible to explain processes or make predictions. The simplest models are in the form of mathematical formulas. The following example of a **linear relation** between two quantities stems from mathematical modelling of human growth and concerns the relation between the height of a girl at menarche and her adult height.

Experimentally it has been found that the height at the time of first menstruation (menarche) is a good indicator of the expected adult height. The following mathematical formula can be found on the website www.medal.org of the Medical Algorithms Project. \[\mathrm{adult\;height\;(cm)} = a\times \mathrm{height\;at\;menarche\; (cm)}+b,\] where the constants \(a\) and \(b\) are taken from the following table. \[\begin{array}{|r|r|r|}\hline

\mathit{age\;at} & & \\

\mathit{menarche\;(year)} & a & b \\ \hline

10 & 0.975 & 15.6 \\

10.5 & 0.976 & 14.7 \\

11 & 0.969 & 14.9 \\

12 & 0.970 & 13.1 \\

12.5 & 0.967 & 13.1 \\

13 & 0.965 & 12.6 \\

13.5 & 0.968 & 11.4 \\

14 & 0.966 & 10.8 \\

14.5 & 0.968 & 9.7 \\

15 & 0.968 & 8.8\\

15.5 & 0.975 & 7.0 \\

16 & 0.977 & 5.8 \\ \hline

\end{array}\] A specific case is the height prediction for girls with a menarche age of 13 year: \[ \mathrm{adult\;height\;(cm)} = 0.965\times \mathrm{height\;at\;menarche\; (cm)}+12.6\]

In the above concrete formula \[\mathrm{adult\;height\;(cm)} = 0.965\times \mathrm{height\;at\;menarche\; (cm)}+12.6\] we call \(\mathrm{height\;at\;menarche}\) the **independent variable,** because it is a variable the value of which can be freely chosen. The \(\mathrm{adult\;height}\) is the **dependent variable** because its value depends on the value of the independent variable.

But even when we use the general formula \[\mathrm{adult\;height} = a\times \mathrm{height\;at\;menarche}+b\] then we still speak of a linear relation between adult height and height at menarche; the variables \(a\) and \(b\) are **parameters**, which are age-related constants in this case. In this example, the relation may thus be regarded as a function, namely \(\mathrm{adult\;height}\) is **a function of** \(\mathrm{height\;at\;menarche}\) .