You now know three families of models: linear, quadratic and exponential. Real data rarely comes labeled, so you need a way to tell which family fits.
Why it works
Over equal steps in $x$: a linear function changes by a constant amount, so its first differences are constant. A quadratic's first differences change, but by a constant amount each time, so its second differences are constant (for $y = x^2$: outputs $0, 1, 4, 9, 16$; first differences $1, 3, 5, 7$; second differences $2, 2, 2$). An exponential function changes by a constant factor, so its ratios are constant. In the long run, exponential growth beats quadratic growth, which beats linear growth.
| $x$ | $0$ | $1$ | $2$ | $3$ | $4$ | Pattern |
|---|---|---|---|---|---|---|
| linear $2x + 1$ | $1$ | $3$ | $5$ | $7$ | $9$ | first differences $2$ |
| quadratic $x^2 + 1$ | $1$ | $2$ | $5$ | $10$ | $17$ | second differences $2$ |
| exponential $2^x$ | $1$ | $2$ | $4$ | $8$ | $16$ | ratios $2$ |
Key idea
Constant first differences: linear. Constant second differences: quadratic. Constant ratios: exponential. Then think about the situation: adding a fixed amount, multiplying by a fixed factor, or rising and falling like a thrown object.
Worked examples
Worked example: Identify the family
Data: $(0, 3)$, $(1, 5)$, $(2, 11)$, $(3, 21)$, $(4, 35)$.
- First differences: $2, 6, 10, 14$. Not constant.
- Second differences: $4, 4, 4$. Constant.
Answer: Quadratic (it is $2x^2 + 3$).
Worked example: The long run
Which grows larger eventually: $50x$ or $1.5^x$?
- At $x = 10$: $500$ vs about $58$. At $x = 20$: $1000$ vs about $3325$.
Answer: $1.5^x$, the exponential.
Common mistake
Fitting any curved data with a quadratic without checking ratios. The data $3, 6, 12, 24, 48$ curves upward, but its ratios are all $2$: it is exponential, and a quadratic would predict far too little later on. Check differences and ratios before choosing.