Definition: Exponential growth occurs when a quantity is multiplied by the same factor greater than 1 in each equal time period, so it increases faster and faster.
What does exponential growth look like?
Linear growth adds the same amount each step. Exponential growth multiplies. A table shows a constant ratio between outputs instead of a constant difference, and the graph bends upward.
What is the model?
y = a(1 + r)t, where a is the starting amount, r is the growth rate written as a decimal, and t is the number of periods. The growth factor 1 + r is greater than 1.
Example. A town of 12,000 people grows 3.5% each year. After 8 years the population is 12,000(1.035)8. Since 1.0358 is about 1.3168, the population is roughly 15,800.
What do students confuse?
Writing 0.035 instead of 1.035 as the base turns growth into a rapid shrink toward zero. Another error is multiplying a by (1 + r) and then by t, which is linear thinking. Also note that the starting amount a is the y-intercept, the value at t = 0.