ACT Math: Exponential Growth and Decay—Model Real-World Scenarios

Exponential growth/decay is modeled by: y=a(b)^t or y=a×e^(kt). Components: a = initial amount. b = growth/decay factor (growth: b>1; decay: 0In exponential growth, the amount increases faster over time; in decay, it decreases but never quite reaches zero (asymptotic).

Key insight: The exponent is time, not the base. If bacteria double every hour, after t hours, you multiply by 2^t, not 2×t. This makes exponential different from linear (which is y=mx+b).

Mistake 1: Confusing growth factor with growth rate. If something doubles, b=2 (growth factor). Growth rate is 100% (0.5 used in certain formulas). Mistake 2: Using wrong exponent. "Population triples every 5 years" doesn't mean y=a(3)^t. It means y=a(3)^(t/5). Time must be scaled to match the period. Mistake 3: Setting up decay incorrectly. Decay factor is between 0 and 1. If something decreases by 30% each year, it retains 70%, so b=0.7, not -0.3 or 0.3. Mistake 4: Misinterpreting "half-life" or "doubling time." Half-life (decay): y=a(0.5)^(t/half-life). Doubling time (growth): y=a(2)^(t/doubling-time). Always check: Is the base >1 (growth) or between 0 and 1 (decay)? Is the exponent adjusted for the time period?

Checklist: (1) Identify a (initial amount). (2) Identify the growth/decay factor or rate. (3) Identify the time period. (4) Set up the equation. (5) Solve for the unkn own (time, amount, or factor). (6) Check reasonableness.

Problem 1: Investment grows at 5% annually. Initial $1000. Value after 10 years? y=1000(1.05)^10≈1629. Problem 2: Bacteria doubles every 2 hours. Start 50 cells. After 6 hours? y=50(2)^(6/2)=50(2)^3=400 cells. Problem 3: Radioactive material has half-life of 8 years. Start 64 grams. After 24 years? y=64(0.5)^(24/8)=64(0.5)^3=8 grams. Problem 4: Population decreases by 10% yearly. Start 10,000. After 5 years? y=10000(0.9)^5≈5905 people. Problem 5: A car depreciates 20% yearly, initially $30,000. Value after 4 years? y=30000(0.8)^4≈9830. For each, identify a, b (or k), t, and verify the exponent is scaled correctly to match the period.

Daily drill: Solve one exponential problem daily, alternating between growth and decay. Practice setting up the equation before solving. Verify answers make sense (growth increases, decay decreases).

About 1-2 questions per ACT Math section involve exponential growth/decay, often in word problem form. These combine real-world context, function setup, and calculation—multiple skills. If you master exponential functions, you answer these efficiently. Many students confuse exponential with linear and set up equations wrong, so correct answers are less common, making these high-value targets for your preparation.

Spend 2-3 days on exponential functions. Drill word problem setup and equation solving. By test day, you'll recognize exponential scenarios, set up functions correctly, and solve confidently.