Integrated Math 2 / Unit 5: Modeling with Functions
5.2 Exponential Growth and Decay
Notes
An exponential function has the form $y = a \cdot b^x$, where $a$ is the starting value and $b$ is the growth factor.
- $b > 1$ is growth. $0 < b < 1$ is decay.
- For a percent rate $r$: growth uses $b = 1 + r$, decay uses $b = 1 - r$.
- Example: $1000 growing 5% per year is $A = 1000(1.05)^t$.
- Example: a $20,000 car losing 10% per year is worth $20000(0.9)^2 = 16200$ dollars after 2 years.
Practice
Which function models exponential decay?
A growth factor between 0 and 1 means the quantity shrinks.
A population of 500 grows 4% per year. About how large is it after 3 years?
$500(1.04)^3 = 500(1.124864) \approx 562$.
A $20,000 car loses 10% of its value each year. What is it worth after 2 years?
$20000(0.9)^2 = 20000(0.81) = 16200$. Losing a flat 10% of the original each year would give $16,000, which is not how percent decay works.
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