Exponential Growth and Decay
A free Algebra I lesson from the “Exponential Functions and Sequences” unit, with a worked example and practice problems including step-by-step solutions.
For exponential growth at rate r per period: y = a * (1 + r)^t. For decay (loss) at rate r per period: y = a * (1 - r)^t. Here a is the starting amount and t is the number of periods. Compound interest, population growth, and depreciation all fit one of these forms. Doubling means multiplying by 2; doubling time is the time needed for that change. If a population doubles every 6 hours, its doubling time is 6 hours and its multiplier over each 6-hour interval is 2.
What you'll learn
- Set up exponential growth models y = a * (1 + r)^t
- Set up exponential decay models y = a * (1 - r)^t
- Apply to compound interest, depreciation, and half-life problems
Worked example
Problem. $1000 invested at 5% annual compound interest. Value after 3 years?
- y = 1000 * (1.05)^3.
- (1.05)^3 = 1.157625.
- y = 1000 * 1.157625 = 1157.625.
- Connect the result back to Exponential Growth and Decay so the method and meaning are both clear.
Answer: 1157.625
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. $500 at 4% annual compound interest. Value after 2 years? (Round to 2 decimal places.)
Show solution
- y = 500 * (1.04)^2 = 500 * 1.0816 = 540.80.
Answer: 540.80
2. A car worth $20,000 depreciates 10% per year. Value after 1 year?
Show solution
- y = 20000 * (1 - 0.10) = 20000 * 0.90 = 18000.
Answer: 18000
3. 200 bacteria double every hour. Count after 3 hours?
Show solution
- y = 200 * 2^3 = 200 * 8 = 1600.
Answer: 1600
4. An 80-gram radioactive sample halves every year. Amount after 3 years?
Show solution
- y = 80 * (1/2)^3 = 80 * 1/8 = 10.
Answer: 10
5. $1000 at 8% annual compound. Value after 2 years? (Round to 2 decimal places.)
Show solution
- y = 1000 * (1.08)^2 = 1000 * 1.1664 = 1166.40.
Answer: 1166.40
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