Key Concepts
- Exponential functions often model real-world phenomena such as population growth, bacteria replication, and financial investments.
- Exponential functions follow the form (f(x) = a \cdot b^x), where a is the initial value and b determines growth or decay.
- If (b > 1), the function models exponential growth (e.g., population increase).
- If (0 < b < 1), the function models exponential decay (e.g., radioactive decay, depreciation).
- Writing an exponential function based on a word problem.
- Many real-world growth models increase by a percentage over time rather than a fixed amount.
- Some exponential models apply changes every $6$ months or every quarter, requiring conversion of the exponent.
Skills Covered
- Writing exponential equations in the form (f(x) = a \cdot b^x)
- Evaluating exponential functions at specific values.
- Writing an exponential growth/decay model given a percentage increase/decrease.
- Interpreting real-world exponential models.
- Adjusting an exponential equation for a non-yearly increase (e.g., every $6$ months).
Example Problems
- A car depreciates in value following (V(t) = 20,000(0.85)^t). Find (V(3)).
- A city's population follows (P(t) = 5000(1.05)^t). What does the $1.05$ represent?
- A population of rabbits grows by 12% per year. If (P(0) = 500), write an exponential model.
- A bacteria culture follows (P(t) = P_0 e^{kt}). Given (P(0) = 100) and (P(5) = 800), find (k).
- A city’s population follows (P(t) = 10,000(1.02)^t), where t is in years. Write an equivalent expression (P(m)) where m is measured in months.