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I have a physics project, and I have to develop an argument, but am not allowed to use phrases like "From the graph you can tell..."

How can the nature of the graph be determined manually, e.g. finding that it is expontential rather than quadratic?

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If you are not allowed to use arguments like “from the graph you can tell …”, you should be given another representation of the function, or else you have no chance. Do you have a concrete mapping assignment? – k.stm May 10 '13 at 7:45
Basically, I have conducted an experiment, and I shouldn't rely on software to develop my argument. – DarkLightA May 10 '13 at 7:57
@DarkLightA So you have a table of data points and wish to draw conclusions from them. – Milo Brandt Mar 8 '15 at 23:31

Well if you guess that it is exponential, you could find a suitable change of variables so that the resulting pair of variables are supposed to be linearly related, and then you can draw a best-fit line and compute the correlation coefficient if appropriate. To test for a quadratic relation, you might want to use some polynomial interpolation technique. You may want to look at and for details.

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To go along with user21820's answer, if you think your data points fit an exponential curve $$ y=ae^{bx} $$ you can use the TI-83/84 to fit an exponential curve to the data. But I am not sure if it will compute any kind of correlation coefficient to help support the idea that an exponential curve is the best fit.

However, you could take logarithms of both sides and get $$ \log y=\log a + b\log x $$ and then you can get a linear fit of your logarithmic data. Then the correlation coefficient of this linear model will tell you how well a line fits your logarithmic data and hence how well an exponential equation fits your original data.

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