Exponential Growth and Decay
Quantities that change by a fixed percent follow exponential models.
About this topic
Exponential models have the form A = P(1 + r)ᵗ for growth and A = P(1 − r)ᵗ for decay, where P is the starting amount, r the rate per period and t the number of periods. A population increasing 5% yearly grows faster each year because the base grows too.
Half-life is the time for a decaying quantity to fall to half. If a medicine has a half-life of 6 hours and you start with 80 mg, you have 40 after 6 hours and 20 after 12, which is A = 80(1/2)^(t/6). A town of 10,000 growing at 3% per year reaches 10,000(1.03)¹⁰, about 13,439, after a decade.
These models describe bacteria, investments, depreciation of a car, carbon-14 dating of fossils, the spread of diseases, and the cooling of objects. Understanding them helps people think clearly about compounding, both good and bad.
Do not convert the percent incorrectly: 5 percent becomes 0.05, so the factor is 1.05, not 1.5. Another error is treating the growth as linear by multiplying the rate by the time. Remember that the same percentage applies to an ever-changing amount, which is why the curve bends upward.
Key ideas
- Growth: y = a(1 + r)ᵗ
- Decay: y = a(1 − r)ᵗ
- Half-life is a decay model
Quick check
Use cases: where Exponential Growth and Decay helps
- Homework: review the key idea behind Exponential Growth and Decay before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Exponential Growth and Decay without notes.
- Visual learning: play with the interactive lab to see how Exponential Growth and Decay changes when you adjust its inputs.
- Teaching a friend: explain Exponential Growth and Decay in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Exponential Growth and Decay?
Quantities that change by a fixed percent follow exponential models.
What will I learn in Exponential Growth and Decay?
Growth: y = a(1 + r)ᵗ. Decay: y = a(1 − r)ᵗ. Half-life is a decay model.
Is Exponential Growth and Decay part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Exponential Growth and Decay lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Exponential Growth and Decay?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.