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Decay Factor

Decay factor is the number you repeatedly multiply by in an exponential decay function. It's the base bb in f(x)=a⋅bxf(x) = a \cdot b^x, and it's always between 0 and 1.

In an exponential function of the form f(x)=a⋅bxf(x) = a \cdot b^x, the decay factor is the base bb when 0<b<10 < b < 1. Each time xx increases by 1, the output is multiplied by bb, causing the function's value to decrease over successive intervals. The decay factor is related to the decay rate rr by the equation b=1−rb = 1 - r, where rr represents the fractional decrease per unit of xx.

Key Formula

b=1−rb = 1 - r
Where:
  • bb = the decay factor (between 0 and 1)
  • rr = the decay rate expressed as a decimal

Worked Example

Problem: A car is worth $20,000 and loses 15% of its value each year. Find the decay factor and write a function for the car's value after t years. Then find its value after 4 years.
Step 1: Identify the decay rate. The car loses 15% per year, so r=0.15r = 0.15.
r=0.15r = 0.15
Step 2: Calculate the decay factor using b=1−rb = 1 - r.
b=1−0.15=0.85b = 1 - 0.15 = 0.85
Step 3: Write the exponential function. The initial value a=20,000a = 20{,}000 and b=0.85b = 0.85.
V(t)=20,000⋅0.85tV(t) = 20{,}000 \cdot 0.85^t
Step 4: Substitute t=4t = 4 to find the value after 4 years.
V(4)=20,000⋅0.854=20,000⋅0.52200625≈10,440.13V(4) = 20{,}000 \cdot 0.85^4 = 20{,}000 \cdot 0.52200625 \approx 10{,}440.13
Answer: The decay factor is 0.85, and the car is worth approximately $10,440.13 after 4 years.

Visualization

Why It Matters

Decay factors appear whenever a quantity shrinks by a consistent percentage — radioactive decay, depreciation of assets, cooling of objects, and the decline of medication in the bloodstream. Understanding the decay factor lets you quickly model how fast something is disappearing and predict future values without recalculating the loss at every step.

Common Mistakes

Mistake: Using the decay rate as the base instead of the decay factor
Correction: If something decreases by 15%, the decay factor is b=1−0.15=0.85b = 1 - 0.15 = 0.85, not 0.150.15. The base must represent what remains, not what is lost.
Mistake: Confusing decay factor with growth factor
Correction: A decay factor is always between 0 and 1. If b>1b > 1, the function is growing, not decaying. Check that your base reflects a decrease.

Related Terms