### Why do this problem?

This
problem offers an opportunity to reflect on the very important
concept of fitting a curve to experimental data. Along the way,
students will utilise their skills of transforming graphs in order
to find a close fit, and consider ways of deciding how close their
fit is. The problem is marked as challenge level 1 as it is a
straightforward task to begin, but to find a complete solution for
all 10 graphs is rather more challenging!

### Possible approach

Although this problem stands alone, it could also be done as
a follow-up to work on transformations of graphs based
on the problem

Parabolic
Patterns.
Students will need access to computers or graphical
calculators to get the best out of this task. Familiarity with
spreadsheet software is assumed.

Part of the challenge of this problem is to identify which
graphs are easiest to fit, as they are not presented in any
particular order. One approach is to start by displaying the graphs
and discussing as a class or in pairs which have recognisable
shapes, such as straight lines, quadratics, trig graphs and
exponential graphs.

If students haven't met graphs such as $y=a^x$ and $y=a^{-x}$
it might be fruitful to give them some time to experiment with
graphical calculators to see what these graphs look like for
different values of the constant $a$.

Once students have some preliminary ideas about graphs which might
fit, small groups could start to work on the spreadsheet, entering
a possible equation and seeing how closely it matches the given
data, then using their knowledge of transformations of graphs to
tweak their equation to get a closer match. Alternatively, they
could experiment with graphical calculators to find graphs with the
right basic shape and then enter them into one copy of the
spreadsheet displayed at the front of the class.

Ideally, different groups will come up with slightly different
suggestions for functions, and this can stimulate discussion about
how to decide which function most closely matches the data.

### Key questions

What clues can we find from the axes and the points given to
help us to guess a likely function?

How can we modify our guess once we've seen how closely it
fits?

Does joining the points in order of increasing time
help?

How do we decide when the fit is close enough?

### Possible extension

Students could investigate and discuss the benefits of a least
squares method of determining how close the fit is.

### Possible support

Graphs 1, 3 and 5 are the most straightforward functions to fit, so
this is a good place to start.