Converting Between Decay Rate and Decay Factor
For exponential decay functions f(x) = a(1-r)ˣ, the decay rate r (as a decimal) and the decay factor (1-r) are directly interconvertible: decay factor = 1 - r and r = 1 - decay factor — a key concept in enVision Algebra 1 Chapter 6 for Grade 11. A car depreciating at 15% per year has r = 0.15 and decay factor 0.85. If a radioactive substance has decay factor 0.92, the decay rate is 1 - 0.92 = 0.08 or 8%. A population declining by 3% annually has r = 0.03 and decay factor 0.97. Correctly identifying whether a percentage represents r or (1-r) prevents errors when building exponential models.
Key Concepts
For exponential decay functions $f(x) = a(1 r)^x$: Decay rate: $r$ (expressed as a decimal) Decay factor: $(1 r)$ Conversion formulas: $r = 1 \text{decay factor}$ and $\text{decay factor} = 1 r$.
Common Questions
What is the decay factor if the decay rate is 15%?
Decay factor = 1 - r = 1 - 0.15 = 0.85. The quantity retains 85% of its value each period.
If a substance has a decay factor of 0.92, what is the decay rate?
r = 1 - 0.92 = 0.08, or 8% per period.
How do you write the exponential decay model for a car depreciating at 15% per year?
f(t) = a(1 - 0.15)ᵗ = a(0.85)ᵗ, where a is the initial value and t is years.
What is the decay factor for a population declining by 3% annually?
Decay factor = 1 - 0.03 = 0.97. Each year the population is multiplied by 0.97.
What is the difference between the decay rate and the decay factor?
The decay rate r tells you what percentage is lost each period. The decay factor (1-r) tells you what fraction remains. For 20% decay: rate = 0.20, factor = 0.80 (80% remains each period).