Equation matcher

Fitting algebraic curves through experimental data points is an important scientific process which allows us to make predictions of the behaviour of a system away from the observed points. But which equations are sensible choices for a particular set of data? This question draws us into the general process of curve fitting.

In four different sorts of experiments, two values of $x$ and $y$ are successively measured and plotted on charts below:

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For each chart, which of the following equations are possible descriptions of the underlying process?

$$y = ax+b\quad\quad x = ay + \frac{1}{2} by^2\quad\quad y=\frac{a}{x}$$

$$x=a e^{by}\quad\quad x = \frac{ay}{b+y}\quad\quad x = -a\log_{10}(by)$$

What restrictions would the data place on the ranges of the numbers $a$ and $b$ in each case? For example, do the data points imply that the constants will be positive?

Another point is subsequently measured and plotted on the graphs in each case.

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How do the possibilities now change?

Can you give examples of physical systems which are modelled by equations of these types?

Extension: Which curves can be distinguished by three points? i. e., could you pick three points which would rule out all but one of the possibilities in each case?