Question: I am conducting research on a plot of data. Most appear linear, as expected, but one series was different. The dark red plot (below) is a more accelerated growth curve - and I'm wondering why?

Here is an image of the plot:
![ ](https://i.stack.imgur.com/jiTqT.png)

* Is this curve exponential?
* What is the best way to describe the rates of change?
* What other ways can we compare the behavior of this curve?

The dataset of the accelerated curve look like this:
["7323", "10376", "14261", "19263", "25890", "34051", "43624", "55430", "70519", "87578", "106268", "127449", "153616"]

Growth readings occurred on these dates:
[7323 2012/02/19, 10376 2012/05/29, 14261 2012/09/06, 19263 2012/12/15, 25890 2013/03/25, 34051 2013/07/03, 43624 2013/10/11, 55430 2014/01/19, 70519 2014/04/29, 87578 2014/08/07, 106268 2014/11/15, 127449 2015/02/23, 153616 2015/06/03]

Answer: 
What you're looking for is called "exponential regression" which is a process which fits your data to the graph of an exponential function, which does in fact, look like what you have as a picture. Furthermore, the process of exponential regression gives a number called a correlation coefficient which is between 0 and 1 which tells you just how well the data fits the exponential curve.

In general, the process of computing these numbers is hard to do by hand because it is very tedious, but easy to do with a computer. So I went and put the data into a calculator I found online which gives the exponential function $$ y= 6768.577 x^{1.2866}$$ with correlation coefficient 0.995, which means that this function fits your data almost exactly.

We conclude that it is *very* likely your curve is an exponential function. 

Edit: Here's a link on regressions: <https://en.wikipedia.org/wiki/Regression_analysis>

And the calculator I used to perform this analysis <http://keisan.casio.com/exec/system/14059930973581>

Almost any graphing calculator can do similar regressions, including regressions with straight lines, quadratic, cubic and sometimes even higher order polynomials as well.
Yes, the dark red curve is at least approximately exponential. You can check this by looking at how far along the horizontal you have to go to double the vertical; it's the same distance each time.

More information about what you're analyzing would help guide further suggestions.