How can I fit a curve to a data set of mostly inflection points?

I have just obtained a data set which is looking at temperature at depth. There are many different temperature profiles at various locations and times. However, in order to save memory, each temperature profile was stored as a temperature at the surface, a temperature at the bottom depth, and temperatures at depths where the concavity of the temperature profile changed. I do have the full temperature profile for a few years of the data, which has allowed me to test how well curve fitting has worked to reproduced the original data, however, I am not too familiar with the various fitting options in the curve fitting toolbox and thus have had no luck. Is there a good way to recreate the temperature profiles when I have end points (which are not inflection points) and data points throughout which are inflection points?

[Matt J]

I don't think it's a question of how many inflection points the curve has. I think it's a question of whether or not you know a good parametric model for the curve.