Integrated Math 3 / Unit 3: Exponential and Logarithmic Functions
3.1 Exponential Functions: Growth and Decay
Notes
An exponential function has the form $f(x) = a \cdot b^x$ with $a \neq 0$, $b > 0$ and $b \neq 1$. Here $a$ is the starting value and $b$ is the growth factor.
- $b > 1$ is growth. $0 < b < 1$ is decay.
- A growth rate of $r$ gives $b = 1 + r$. A decay rate of $r$ gives $b = 1 - r$.
- Example: a population of 500 growing 4% per year is $P(t) = 500(1.04)^t$.
- The graph has a horizontal asymptote ($y = 0$ for $a \cdot b^x$) and $y$-intercept $a$.
Practice
What is $f(3)$ for $f(x) = 2(5)^x$?
$2 \cdot 5^3 = 2 \cdot 125 = 250$.
A car worth $24,000 loses 15% of its value each year. Which function gives its value after $t$ years?
Losing 15% each year means keeping 85%, so the growth factor is $0.85$.
Which function shows exponential decay?
Decay requires the base to be between 0 and 1.
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