Ok they’re not graphs of temperature but temperature anomaly, which is the difference between temperature at average over some period.
A combination of 2 cosines
A straight line along average temperature
The amplitude, period and phase of 2 cosines were free parameters.
Initial values for the periods were 50 and 500 years, final values were about 64 and 305.
The two cosines were constrained to have a common peak at some point, with the initial value set at 1992, but this ended up at 2097! I had expected 1998. The extended curve looks like.
With four parameters I can fit an elephant, with five I can make him wiggle his trunk
[Digression John D Cook shows how to draw an elephant with 4 parameters in python]
Trying to predict how a curve will develop outside the range show on a graph can give results which are completely wrong. You should only attempt this if you know (or at least a strong reason for believing) what the equation which describes the curve is.
Trick question – the graph actually had 6 different curves on which develop like this
And here are the curves over a wider range
The data and python script used to produce the temperature graphs are here