The Tortoise and the Hare

Learning Focus

Compare quadratic and exponential growth using tables, rates, and long-run behavior.

Big idea

• Quadratic, e.g. $q(n)=n^2$: constant second difference; growth by adding increasing amounts.
• Exponential, e.g. $e(n)=2^n$: constant ratio; growth by multiplying.

The quadratic may lead early, but an exponential with base greater than 1 eventually overtakes every quadratic.

| $n$ | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | $n^2$ | 1 | 4 | 9 | 16 | 25 | 36 | | $2^n$ | 2 | 4 | 8 | 16 | 32 | 64 |

Worked example

Table values $3,6,12,24$: ratios are all $2$, differences ($3,6,12$) are not constant. This is exponential: $y=3\cdot 2^{x}$ (starting at $x=0$).

Takeaways

• Check ratios for exponential, second differences for quadratic.
• Exponential growth eventually beats any quadratic.
• Early values can be misleading; compare the long run.