A process where a quantity increases by a fixed percentage over equal time intervals.
A process where a quantity decreases by a fixed percentage over equal time intervals.
Exponential functions are used to model real-world scenarios involving growth or decay. These functions have the form:
$$y = a \cdot b^x$$
Where:
- $a$ is the initial value.
- $b$ is the growth or decay factor.
- $x$ is the independent variable (often time).
If $b > 1$, the function models exponential growth. If $0 < b < 1$, it models exponential decay.
Using Exponential Equations
Exponential Growth
Population Growth
The population of a city grows by 5% each year. If the initial population is 100,000, the population after $t$ years is given by:
$$P(t) = 100,000 \cdot (1.05)^t$$
1. Can you create an exponential model for a scenario not covered here? Try modeling the spread of a virus or the cooling of a hot beverage. 2. How would you explain the difference between exponential growth and decay to someone unfamiliar with mathematics? 3. Can you think of a real-world scenario where a logarithmic model would be more appropriate than an exponential one?
How do we decide which mathematical model best fits a real-world scenario? What are the limitations of using exponential and logarithmic models in real life?
