1.5: Exponential functions.

Main Points
The exponential function was defined as being in the form f(x) = a^x, with a being a positive constant. With exponential functions the dependent value increases faster per the independent values and have a constant ratio. A typical example of a exponential function is a population growth.
Another form for the exponential function is P = Poa^t where Po is the initial amount at t = 0. Exponential growth occurs when a > 1 and decay occurs when 0 < a < 1. Also considering "a" is a factor of P, the bigger "a" is the faster P grows or decays.
For both a growth (increase) and decay (decrease) the graphs of the functions both are concave up. The difference between the linear and exponential functions is that the linear functions have an absolute rate of change whilst the exponential function has a percentage rate of change.
e is the most commonly used base.

Challenges
A decay always has the x-axis as its asymptote because it never gets to zero and even if it could it could not go beyond it because it cannot go into the negatives. But this is not the same for a growth, it has no asymptote because there are an infinite number of numbers and therefore P can increase forever. This still does not make sense to me though...

Reflections
Whatever medication you take technically never leaves your bloodstream because it never decays to a value of 0! The fact it never gets to zero is possibly the theory behind immunizations. This also explains why there is always a certain amount of time that must pass between two doses of a certain medication. If this time frame was not observed an overdose is more likely to occur. This is good to know, I nearly took my anti-malaria pill 24hrs too early.