High School: Functions
Linear, Quadratic, and Exponential Models HSF-LE.A.3
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Students should be able to prove that eventually, as long as the functions are headed in the same direction, a quantity increasing exponentially will "beat" linear, quadratic, and polynomial functions. Not much to it.
It's probably obvious that the function y = 3x will eventually surpass y = 3x + 3. We can see this via a table of values or a graph. Somewhere down the line, when x gets closer and closer to infinity, the y value of the exponential function will be larger than the y value of the linear function.
We can see that this happens at x = 2 whether we graph it or look at the table of values.
|x||3x||3x + 3|
What about other functions? Ones with exponents that aren't 1 or x? What about something like y = x1000 compared to y = 1000x? At a large enough x, will 1000x really surpass x1000?
The short answer is that yes, it will. Once x = 1000, the two will be equal. For anything greater, the exponential function will emerge victorious. Because even when x = 1001, we know that 10001001 > 10011000. Eventually, any exponential function with a base greater than 1 will override polynomial functions.