Building an Exponential Model

A line adds the same amount every step. A dish of bacteria does not work that way: each new generation multiplies the population, so the more bacteria you have, the more new bacteria you get per hour. That is repeated multiplication, and repeated multiplication is exponential, not linear. The curve bends upward because growth feeds on itself.

To build the model, convert the percent change into a growth factor. A 7 percent yearly increase means each year's amount is 107 percent of the last, so the factor is 1 + 0.07 = 1.07. A percent decrease works the same way with subtraction: a 7 percent decay gives a factor of 0.93. Once you have the factor b, the model is P(t) = P0 x b^t, where P0 is the starting amount and t counts the elapsed periods. For a town of 500 people growing 3 percent a year, P(t) = 500(1.03)^t, and P(25) = 500(1.03)²⁵, which comes out to about 1047 people.

This model assumes growth heads toward 0 or runs unchecked forever, which is fine for short-term population or interest problems but wrong once a real process levels off at some nonzero floor, like a cooling cup of coffee settling at room temperature. When the data flattens at a value c instead of 0, you subtract c from every output first, fit the plain exponential a x b^x to what remains, then add c back as +k in the final model. That adjustment is the bridge into logistic and shifted exponential models later in the unit.