Exponential growth is identified in tables by a constant multiplicative ratio between successive outputs for equal increases in the input, and in graphs by an increasing, increasingly steep curve consistent with , where and , rather than the constant additive difference of linear change. The learner interprets as the growth factor and corresponding percent change over each equal interval, distinguishing multiplicative from additive change; logarithmic methods and more advanced parameter cases are outside this scope.
To recognize exponential growth, check how the output changes when the input increases by equal amounts.
A table shows the population of bacteria:
| Time, (hours) | Population, |
|---|---|
Step 1: Check the differences.
The differences are not constant, so the table is not showing linear growth.
Step 2: Check the ratios.
The ratio is constant. Each output is multiplied by when the time increases by hour. Therefore, the population shows exponential growth.
The function is
Here:
To find the percent increase, subtract from the growth factor:
So the population increases by 50% each hour.
On a graph, the points would form an increasing curve that becomes steeper as increases. This upward-bending shape is consistent with exponential growth, unlike a straight line with a constant slope.
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