In the description of trendlines you can read: 
”Unlike linear trendlines, there are several different ways to compute 
exponential trendlines. We provide only one method right now, but will 
support more in the future, and so it's possible that the name or behavior 
of the current exponential trendline will change.” 
(https://developers.google.com/chart/interactive/docs/gallery/trendlines ).

This text has been there for quite a while and I really would like to know 
the timeframe for when “more” is coming? 

I am also very interested in getting a function that retrieves the R2 value.

I now use the trendline functionality in forecast calculations and think 
the trendline finctionallity is awesome and would be so even more if more 
functionality is added to it.

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