Exponential Growth & Decay
Exponential growth and decay models help students understand how quantities change at rates proportional to their current value. From bacterial colonies that double every hour to radioactive substances losing half their mass over fixed time periods, these mathematical patterns appear throughout science and finance.
Why it matters
Students encounter exponential functions in biology when studying population growth, where bacteria colonies can expand from 100 to 6,400 organisms in just 6 hours through doubling. In chemistry, radioactive decay problems show how 240 grams of carbon-14 reduces to 60 grams over 11,460 years. Financial literacy depends on understanding compound interest, where $5,000 invested at 8% annual growth becomes $10,794 after 10 years. Medical professionals use half-life calculations to determine drug dosages, knowing that 400mg of a medication might decrease to 100mg after 8 hours. Climate scientists model atmospheric carbon dioxide increases of 2.5% annually to predict environmental changes. These real-world applications make exponential functions essential for students pursuing STEM careers and informed citizenship.
How to solve exponential growth & decay
Exponential Growth
- General form: y = a · bˣ, where a is the starting value and b is the growth factor.
- If b > 1, the quantity grows; if 0 < b < 1, it decays.
- Percent growth of r% means b = 1 + r/100.
- To find y after x periods, substitute and evaluate.
Example: A population of 500 grows 10% per year. After 3 years: y = 500 · 1.10³ ≈ 665.5.
Worked examples
A bacteria colony starts with 50 bacteria and doubles every hour. How many bacteria are there after 3 hours?
Answer: 400
- Identify the doubling pattern → 50 × 2³ — The colony doubles 3 times, so multiply by 2³.
- Calculate the power → 2³ = 8 — 2 multiplied by itself 3 times is 8.
- Multiply by the starting amount → 50 × 8 = 400 — There are 400 bacteria after 3 hours.
A town has 8,000 people and grows by 25% per year. How many people live there after 3 years?
Answer: 15625
- Find the growth factor → 1 + 25/100 = 1.25 — A 25% increase means multiplying by 1.25 each year.
- Year 1 → 8000 × 1.25 = 10000 — After year 1 the population is 10000.
- Year 2 → 10000 × 1.25 = 12500 — After year 2 the population is 12500.
- Year 3 → 12500 × 1.25 = 15625 — After year 3 the population is 15625.
- Verify with formula → A = 8000 × 1.25³ = 15625 — Using A = P × (1 + r)ᵗ confirms the answer.
A radioactive sample of 80 g has a half-life of 6 years. How much remains after 12 years?
Answer: 20 g
- Find number of half-lives → 12 ÷ 6 = 2 — In 12 years, the sample halves 2 times.
- Halving 1 → 80 ÷ 2 = 40 — After 6 years: 40 g remaining.
- Halving 2 → 40 ÷ 2 = 20 — After 12 years: 20 g remaining.
- Verify with formula → 80 × (1/2)² = 20 — Using the half-life formula confirms the answer.
Common mistakes
- Students often confuse exponential and linear growth, writing 50 + 2×3 = 56 instead of 50 × 2³ = 400 when bacteria double every hour for 3 hours.
- Many students calculate percentage growth incorrectly, using 8000 × 0.25³ = 125 instead of 8000 × 1.25³ = 15,625 for 25% annual growth over 3 years.
- Students frequently add half-lives instead of applying them sequentially, calculating 80 - 6 - 6 = 68 grams instead of 80 × (1/2)² = 20 grams after two half-life periods.
- When finding growth rates, students often forget to convert decimals to percentages, stating the rate as 0.15 instead of 15% from a growth factor of 1.15.