Algebra 2 / Unit 5: Exponential and Logarithmic Functions
5.1 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.
- $b > 1$: exponential growth. $0 < b < 1$: exponential decay.
- With a percent rate $r$: growth $y = a(1 + r)^t$, decay $y = a(1 - r)^t$.
- Example: a $200 item that loses 10% per year is worth $200(0.9)^t$ dollars after $t$ years.
- Half-life: $\displaystyle y = a\left(\frac{1}{2}\right)^{t/h}$, where $h$ is the half-life.
Practice
The function $y = 500(0.85)^t$ models:
$0.85 = 1 - 0.15$, so it shrinks by 15% each year.
A town of 2,000 people grows 3% per year. About how many people live there after 10 years?
$2000(1.03)^{10} \approx 2687.8$. Using $2000(1.3)^{10}$ treats 3% as 30%.
A sample has 80 grams of a substance with a half-life of 5 days. How much is left after 20 days?
20 days is 4 half-lives: $\displaystyle 80\left(\frac{1}{2}\right)^4 = 5$.
Ready for more? Take the Unit 5 test.