We now know that any time an [i]initial value[/i] increases or decreases by the [i]same factor[/i] or percentage over equal increments of time, we can use an exponential model.[br][br]The factor or percentage is also known as [i][b]relative growth[/b][/i]: we mostly see it expressed as a [i]percentage [/i]in real life models. [br]For example, if we read that a population of crabs grows by 1.2% every year, this means that for every 1000 crabs in the population, other [math]1000\cdot\frac{1.2}{100}=1000\cdot0.012=12[/math] crabs will be added to the population every year.

We can further refine the general model for an exponential growth [math]f\left(x\right)=ab^x[/math] to an equivalent form that shows visually the main characteristics of the model, and precisely:[br][br][center][math]f\left(x\right)=a\left(1+r\right)^x[/math][/center]where:[br][list][*][math]a[/math] is the initial value[br][/*][*][math]r[/math] is the relative growth rate (percentage of growth rate), expressed as a positive decimal number[/*][/list][br]A typical example is the model for populations growth, that is a function of [i]time[/i], usually written as [math]P\left(t\right)=P_0\left(1+r\right)^t[/math] where:[br][list][*][math]P_0[/math] is the initial population[br][/*][*][math]r[/math] is the relative growth rate (percentage of growth rate), expressed as a positive decimal number[/*][*][math]t[/math] is the time unit[br][/*][/list]

Use the applet below to get familiar with exponential growth.[br]Enter your answers (press the [i]Enter [/i]key after each entry), then select [i]Check answers[/i] to get a visual grading of your answers. [br]If they are both correct, a button will allow you to create and explore a new model, otherwise you can deselect [i]Check answers[/i] and try to enter the correct ones.[br]Select the [i]Show solution[/i] checkbox to view the solution of the problem.

If we want to model an exponential decay, the equation will be instead:[br][br][center][math]f\left(x\right)=a\left(1-r\right)^x[/math][/center][br]Therefore, a population that is exponentially decreasing over time will be modeled by a function of the form [math]P\left(t\right)=P_0\left(1-r\right)^t[/math].[br]All the variables are defined as above for the exponential growth function.

Use the applet below to get familiar with exponential decay.[br]Enter your answers (press the [i]Enter [/i]key after each entry), then select [i]Check answers[/i] to get a visual grading of your answers. [br]If they are both correct, a button will allow you to create and explore a new model, otherwise you can deselect [i]Check answers[/i] and try to enter the correct ones.[br]Select the [i]Show solution[/i] checkbox to view the solution of the problem.