Exponential growth and decay
Every model in this topic has the same skeleton: a starting amount times a factor raised to a power. The questions then ask what one piece of that skeleton means in the story. Read the skeleton once and the questions stop looking different.
The skeleton
Write it as a times b to the t. The a is the amount at time zero, so it is the starting population, the initial dose, the purchase price. The b is the factor the amount is multiplied by each period, and it is always positive. The t counts periods, so it is usually time, but it can also be the number of cycles.
The base carries the percent
A base above 1 grows; a base below 1 decays. The percent change per period is the base minus 1, read as a signed percentage. So 1.08 is a growth of 8 percent per period, 0.92 is a decrease of 8 percent, 2 is doubling, and 0.5 is halving. This one translation answers most of the questions in the topic, including the ones phrased as by what percent.
- 1.05 means up 5 percent per period
- 0.85 means down 15 percent per period
- 1.5 means up 50 percent per period, not 1.5 percent
- 0.5 means halving, not down 50 percent per year unless the period is a year
When the period is not one unit
If a quantity doubles every 5 hours, the model is 2 to the power t over 5. The exponent is counting how many five hour periods have passed. A question asking how many cells exist after 20 hours is really asking for the exponent 20 divided by 5, which is 4, so the amount is multiplied by 2 four times. Writing the exponent as a division is the step students skip, and it is the step the question is testing.
Finding the pieces from data
The starting value is the entry at t equals zero, which is often the first row of a table. The base comes from the ratio of one period to the next: divide consecutive values and the base appears. If the values are 500, 600, 720, each step is a multiplication by 1.2. When two points are given as coordinates, divide the later value by the earlier one and count how many periods separate them.
Reading what is actually asked
Three phrasings cover the topic, and they want different numbers. By what factor wants the base. By what percent wants the base minus 1. How many remain after a given time wants the whole model evaluated. Answer the first two with the base and the third with the output, and do not convert between them unless the question forces it.
Half-life in one line
Halving means multiplying by one half, so a half-life model is a times one half to the power t over h, where h is the half-life. To find h from data, count how many halvings separate the starting amount from the final amount, then divide the elapsed time by that count. From 640 down to 40 is four halvings, so if that took 32 hours the half-life is 8 hours.
Is a base of 0.8 a 20 percent decrease?
Yes. The base minus 1 is -0.2, which is a decrease of 20 percent per period.
What does the number in front of the base mean?
It is the value at time zero, before any growth or decay. In a population model of 500 times 3 to the t, the population started at 500.
How do I find the growth factor from a table?
Divide each value by the one before it. If the ratio is constant, that ratio is the base, and a constant ratio is what makes the data exponential rather than linear.
How do I turn a percent into a model?
Add or subtract the rate from 1. A 6 percent annual increase starts at 1.06, and a 6 percent annual decrease starts at 0.94.
Related guides
Updated 2026-10-25. Free to use, no account needed. SAT is a trademark of College Board. Not affiliated with or endorsed by College Board.