Decay Factor in Exponential Functions Explained
What the Decay Factor Actually Is
The decay factor is the base number less than 1 in an exponential decay function. It tells you what portion of the previous value remains after each time interval.
Think of it as the multiplier that shrinks your quantity step by step. If your decay factor is 0.7, you're keeping 70% of what you had before each step.
The Formula
Exponential decay follows this pattern:
y = a(1 - r)t
Where:
- a = the starting amount
- r = the decay rate (as a decimal)
- t = time elapsed
- (1 - r) = the decay factor
Some textbooks write it as y = a(b)t where b is the decay factor directly. When b is between 0 and 1, you have decay. When b is greater than 1, you have growth.
Decay Factor vs Growth Factor
This trips people up constantly. Here's the difference:
- Decay factor: a number between 0 and 1 (example: 0.85)
- Growth factor: a number greater than 1 (example: 1.15)
A decay factor of 0.85 means you lose 15% each period. A growth factor of 1.15 means you gain 15% each period.
How to Find the Decay Factor
Method 1: From the Percent Decrease
If you know the percentage decrease per period, subtract from 1:
Decay Factor = 1 - (percentage decrease ÷ 100)
Losing 20% per year? Decay factor = 1 - 0.20 = 0.80
Method 2: From Two Data Points
Divide the later value by the earlier value:
Decay Factor = (later amount ÷ original amount)
Something drops from 100 to 85 in one period? 85 ÷ 100 = 0.85
Method 3: From the Formula
If your function is y = 500(0.7)t, the decay factor is 0.7. That's it. Whatever number is raised to the power of t, assuming it's less than 1.
Practical Example: Carbon Dating
Carbon-14 decays at about 0.012% per year. The decay factor is:
1 - 0.00012 = 0.99988
That tiny difference between 1 and the decay factor is why carbon dating works over thousands of years. Each year you have 99.988% of what you had before.
After 5,730 years (one half-life), your decay factor compounded gives you roughly 50% remaining. That's why archaeologists can date ancient materials.
Common Decay Factor Values
| Percent Lost Per Period | Decay Factor | After 3 Periods |
|---|---|---|
| 10% | 0.90 | 0.729 (72.9% left) |
| 25% | 0.75 | 0.422 (42.2% left) |
| 50% | 0.50 | 0.125 (12.5% left) |
| 75% | 0.25 | 0.016 (1.6% left) |
Where Decay Factors Appear
- Radioactive decay: isotopes losing stability over time
- Drug metabolization: half-life of medications in your bloodstream
- Depreciation: cars losing value the moment you drive them off the lot
- Cooling: Newton's Law of Cooling uses a decay factor approaching room temperature
- Population decline: areas losing residents year over year
- Skill decay: unused knowledge fading from memory
How To: Calculate Remaining Amount Using Decay Factor
Problem: A substance starts at 200 grams and decays at 12% per hour. How much remains after 5 hours?
Step 1: Find the decay factor
1 - 0.12 = 0.88
Step 2: Set up the equation
y = 200(0.88)5
Step 3: Calculate
0.885 = 0.88 × 0.88 × 0.88 × 0.88 × 0.88 = 0.5277
y = 200 × 0.5277 = 105.54 grams
After 5 hours, about 106 grams remain.
Why Decay Factors Matter
Once you understand decay factors, you can predict outcomes that simple subtraction gets completely wrong. Linear thinking says 12% off five times is 60% gone. Exponential thinking with a decay factor shows you're still keeping 53%.
This distinction matters for financial decisions, scientific analysis, and any scenario where the rate of change itself is changing.
Quick Reference
- Decay factor is always between 0 and 1
- Smaller decay factor = faster decay
- Decay factor of 0.5 = half-life (50% remains after one period)
- To find decay factor from a percentage: 1 - r
- To find percentage decay: (1 - decay factor) × 100