When a quantity increases by a set amount for each unit time, it is called a linear growth model, and obeys the rules covered in
chapter 3 through
chapter 5. That set amount is the slope. But in some cases the
amount the quantity increases isn't constant, instead it is a set percent. In this case it is an exponential growth model. The general
form for an exponential growth model is:
y = C(1 + r)
t ; where C is the initial amount, r is the percent that it grows by, and t is how many times it has grown.
The most common example of exponential growth in the real world is interest gained in a bank. If you have $1000 and the account gains 5% interest
per year you would express the current balance as 1000(1 + 0.05)
t; where t is the number of years. If t is 0, no time has passed, the
(1 + 0.05)
0 reduces to 1 (remember, anything to the 0 power is 1), and you get 1000 which is what you started with. If t = 1, a single
year has passed, which means you should have gained 5%, or $50. The equation shows this to be true: 1000(1 + 0.05)
1 = 1000(1.05) = 1050.
If two years pass, you would expect to gain 5% of the 1050, or $52.50, for a total of $1102.50. Plugging into the equation verifies ths:
1000(1 + 0.05)
2 = 1000(1.05)
2 = 1000(1.1025) = $1102.50.