Linear, Quadratic or Exponential?

With the $x$-values equally spaced, run three tests on the $y$-values. First differences: subtract each $y$-value from the next one. If they are all the same, the function is linear. Second differences: subtract consecutive first differences. If those are all the same (and the first differences were not), the function is quadratic. Ratios: divide each $y$-value by the one before it. If those are all the same, the function is exponential. Always subtract and divide in the same direction, later value minus (or divided by) earlier value, so the signs mean something: a negative first difference means the values are falling, and a ratio between 0 and 1 means decay. Check that the $x$-values really are equally spaced before you start, and if none of the three columns is constant, the data fit none of the three types exactly. Real measurements often land there, so look for the test that is nearly constant.

The tests are not rules of thumb; each follows from the algebra. For a line $y=mx+k$, moving one step right always adds $m$, so the first differences are all $m$. For a quadratic $y=a{x}^2+bx+c$, the first difference from $x$ to $x+1$ works out to $2ax+a+b$. That is itself a linear expression in $x$, so the first differences go up by the same amount each time and the second differences are all $2a$: halve the second difference to get the leading coefficient. For an exponential $y=a\cdot b^x$, the product law gives $a\cdot b^{x+1}=a\cdot b^x\cdot b$, so each step multiplies by $b$ and the ratios are all $b$. Now look at the exponential's first differences. Factoring out $a\cdot b^x$ (the formula below) shows that each difference is the current $y$-value times the constant $(b-1)$, so the differences form an exponential with the same ratio $b$. Take differences again and you get yet another exponential with ratio $b$, forever. That is the tell-tale sign of an exponential in a difference table: no column of differences ever settles down, but every column grows by the same factor as the $y$-values. In the figure, $y=2^{x+1}$ has $y$-values $2,4,8,\dots$ and first differences $2,4,8,\dots$, all doubling. (With an $x$-step of $h$ instead of 1, the constants become $mh$, $2a{h}^2$ and $b^h$.)

Measurements are not always taken at even intervals, and when they are not, raw differences and ratios lie. The line $y=2x+1$ sampled at $x=0,1,3,4,7$ gives $y=1,3,7,9,15$, and the first differences $2,4,2,6$ look anything but constant. The fix is to compare each change with the size of its step. For a line, divide each $\Delta y$ by its $\Delta x$: $\frac{2}{1},\frac{4}{2},\frac{2}{1},\frac{6}{3}$ are all 2, so it really is linear with slope 2. For an exponential, a ratio across a step of $h$ is $b^h$, so take the $h$-th root to get the factor per unit step.

Real measurements contain noise, so the ratios of genuinely exponential data will not match exactly. A bacteria culture counted every hour gave $520,790,1190,1820,2730,4150\text{.}$ The ratios are about $1.52,1.51,1.53,1.50,1.52$. They wobble, but they wobble around one value, with no upward or downward trend, which is what exponential data with measurement error looks like. Compare a quadratic: $y=x^2+x+2$ gives ratios $2,2,1.75,1.57,1.45$, which drift steadily toward 1. A steady drift is a pattern, not noise, and it says "not exponential". In the same way, look at all three tests and ask which one is closest to constant: here the first differences $270,400,630,910,1420$ and the second differences $130,230,280,510$ both keep climbing, so exponential is the clear best choice.

Any exponential with $b>1$ eventually passes any linear or quadratic function, however large their coefficients. The ratio test explains why. Each step right multiplies $2^x$ by 2, always. Each step multiplies $100x^2$ by ${\left(\frac{x+1}{x}\right)}^2$, which is 1.21 at $x=10$, about 1.14 at $x=15$, and keeps shrinking toward 1. A fixed factor of 2 beats a factor that is sinking toward 1, so the exponential must catch up and then pull away. Concretely, $2^x$ is far behind at first ($2^{10}=1024$ against $100{(10)}^2=10000$), but it first passes $1000x$ at $x=14$ and $100x^2$ at $x=15$, and never falls behind again. The same reasoning ranks a quadratic ahead of a linear function in the long run: $100x^2$ passes $1000x$ for every $x>10$. A decaying exponential ($0<b<1$) is the exception: it heads toward its asymptote and is eventually passed by any increasing line.