🌆 Grade 9 · Lesson 11 of 12
Decay Lab: Half-Lives and Horizontal Asymptotes
Cut a quantity in half, then in half again — it keeps shrinking forever, always approaching zero but never quite touching it.
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The big idea: Exponential decay multiplies a quantity by a fixed factor smaller than 1 each time period, so it shrinks toward zero without ever actually reaching it. The line it approaches but never touches is called a horizontal asymptote. Half-life describes decay in a very natural unit: the fixed amount of time it takes for the quantity to cut itself exactly in half.
Shrinking, forever
A dose of medicine leaves the bloodstream a little at a time. A radioactive sample loses a fraction of its atoms as time passes. Both are examples of exponential decay: a quantity that is multiplied by the same shrinking factor over and over, getting smaller and smaller but never disappearing completely.
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Decay multiplies by a factor less than 1
In exponential decay, y = a · b^t with 0 < b < 1. Each time period, the quantity is multiplied by b, shrinking it toward zero. Compare this to growth, where b is greater than 1.
\[ y = a\left(\dfrac{1}{2}\right)^{t/h} \]
Half-life form: a is the starting amount, h is the half-life (time to halve), and t is elapsed time. Every h units of time, the quantity is cut exactly in half.
🎮 Decay Lab LIVE
Watch a quantity halve each half-life and approach — but never touch — its horizontal asymptote.
Half-life: cutting in half, again and again
A substance's half-life is the fixed amount of time it takes for exactly half of it to remain. Whatever amount you start with, after one half-life you have half of it; after two half-lives you have half of that (a quarter of the original); after three, an eighth — the fraction remaining is always (1⁄2) raised to the number of half-lives that have passed.
📝 Worked example: An 80 mg dose of medicine has a half-life of 4 hours. How much remains after 12 hours?
- 12 hours is 12 ÷ 4 = 3 half-lives.
- Remaining fraction = (1⁄2)³ = 1⁄8.
- Remaining amount = 80 × (1⁄8).
✓ <strong>10 mg</strong> remain after 12 hours.
📝 Worked example: y = 200(0.9)^t models a cooling effect. What is y when t = 5?
- Substitute t = 5: y = 200(0.9)^5.
- Compute (0.9)^5 ≈ 0.590.
- Multiply: y ≈ 200 × 0.590.
✓ y ≈ <strong>118.1</strong> after 5 time periods.
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The horizontal asymptote is the floor it never reaches
As t grows larger and larger, y = a · b^t (with 0 < b < 1) gets closer and closer to 0, but never actually equals 0. That line, y = 0, is the function's horizontal asymptote — a boundary the graph approaches endlessly without ever touching.
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Decay never truly hits zero
It is tempting to think a decaying quantity eventually runs out completely in a finite amount of time. Mathematically, exponential decay only ever gets closer to zero — arbitrarily small, but never exactly zero — no matter how many half-lives pass.
Check your understanding
1. A substance has a half-life of 5 years. Starting with 40 g, how much remains after 15 years?
15 years is 3 half-lives, so the remaining fraction is (1/2)³ = 1/8. 40 × 1/8 = 5 g.
2. For y = 100(0.8)^t, what value does y approach as t gets very large?
Since 0.8 < 1, repeated multiplication shrinks the quantity toward its horizontal asymptote at y = 0.
3. Which value of b represents exponential decay in y = a·b^t?
Decay requires a growth factor strictly between 0 and 1, so b = 0.5 shrinks the quantity each period.
4. After how many half-lives is a quantity reduced to 1/8 of its original amount?
(1/2)³ = 1/8, so it takes exactly 3 half-lives.
5. Does exponential decay ever reach exactly zero?
Exponential decay approaches its horizontal asymptote at y = 0 but, mathematically, never actually reaches it.
✅ Key takeaways
- Exponential decay multiplies a quantity by a fixed factor b, where 0 < b < 1, shrinking it each period.
- Half-life is the fixed time it takes a quantity to reduce to exactly half its previous amount.
- After n half-lives, the remaining fraction is (1/2)ⁿ.
- The horizontal asymptote y = 0 is the boundary a decay curve approaches endlessly but never touches.
- Mathematically, exponential decay never reaches exactly zero, no matter how much time passes.