Exponential growth and decay
Some quantities grow or shrink by the same percent every period. What single idea describes all of them, and how far can it be trusted?
Where you are 6 inside
What this area is about
A savings balance, a medicine dose in the blood, a phone’s resale value: each changes by the same percent every period, so it is multiplied by the same number each time. That one multiplier, rather than a fixed amount added, is what makes change exponential. This area builds rules of the form y = a \cdot b^x (with b > 0 and b \ne 1), reads them, and compares them with other kinds of growth.
How these fit together
The core idea. Growth and decay factors name the multiplier b: above 1 for growth, between 0 and 1 for decay. Doubling time and half life turn that factor into one length of time.
A procedure. Building a model from two points finds both a and b from two readings.
Comparisons. Linear versus exponential growth races equal differences against equal factors, and linear, polynomial and exponential growth compared adds powers such as x^2 to the race.
A ceiling. Logistic growth grows almost exponentially at first, then levels off at a carrying capacity.
A path through it
Start with growth and decay factors, then doubling time and half life, then the model from two points. Race a line against an exponential before the wider comparison with polynomials. Logistic growth comes last, after you can build an exponential model.
Next come graphs of exponential functions, compound interest and solving exponential equations.