Integrated Math 1 / Unit 5: Exponential Functions and Geometric Sequences
5.3 Exponential Decay
Notes
In exponential decay the factor is between 0 and 1. A decrease of $r$ gives $b = 1 - r$, so 15% decay means $b = 0.85$.
- Model: $y = a(1 - r)^t$.
- Half-life is the time for a quantity to drop to half.
- Example: 160 mg with half-life 3 hours. After 9 hours (3 half-lives): $\displaystyle 160 \cdot \left(\frac{1}{2}\right)^3 = 20$ mg.
- The graph decreases but never reaches 0.
Practice
A car worth $20,000 loses 15% of its value each year. Which function gives its value after $t$ years?
Losing 15% leaves 85%, so the factor is 0.85.
By what percent does $y = 80(0.6)^x$ decrease with each step in $x$?
Each step keeps 60% of the value, so it loses 40%.
A 160 mg dose of a medicine has a half-life of 3 hours. How much remains after 9 hours?
9 hours is 3 half-lives: $160 \to 80 \to 40 \to 20$.
Ready for more? Take the Unit 5 test.