Real-World Examples of Exponential Functions (Algebra 2 Applications)

📚 Quick Study Guide

• 📈 Definition: An exponential function has the form $f(x) = ab^x$, where $a$ is the initial value, $b$ is the growth/decay factor, and $x$ is the independent variable.
• 🌱 Exponential Growth: Occurs when $b > 1$. The function increases rapidly as $x$ increases. Think population growth or compound interest.
• 🍂 Exponential Decay: Occurs when $0 < b < 1$. The function decreases rapidly as $x$ increases. Think radioactive decay or depreciation.
• 🏦 Compound Interest: A common application. The formula is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the time in years.
• ☢️ Half-Life: The time it takes for half of a substance to decay. The formula is $N(t) = N_0(\frac{1}{2})^{\frac{t}{T}}$, where $N(t)$ is the amount remaining after time $t$, $N_0$ is the initial amount, and $T$ is the half-life.
• 🦠 Bacterial Growth: Often modeled by exponential functions, where the growth rate is proportional to the current population size.

🧪 Practice Quiz

1. What is the initial value in the exponential function $f(x) = 5(2)^x$?
   1. A. 2
   2. B. 5
   3. C. 10
   4. D. 0
2. A population of bacteria doubles every hour. If you start with 100 bacteria, how many will you have after 3 hours?
   1. A. 300
   2. B. 600
   3. C. 800
   4. D. 1200
3. A car depreciates at a rate of 15% per year. If the car initially cost $20,000, what is its value after 5 years?
   1. A. $6,000
   2. B. $8,874
   3. C. $10,000
   4. D. $13,000
4. Radium-226 has a half-life of 1600 years. If you start with 50 grams, how much will remain after 4800 years?
   1. A. 6.25 grams
   2. B. 12.5 grams
   3. C. 25 grams
   4. D. 37.5 grams
5. Which of the following functions represents exponential decay?
   1. A. $f(x) = 3(1.2)^x$
   2. B. $f(x) = 5(0.8)^x$
   3. C. $f(x) = 2x + 1$
   4. D. $f(x) = x^2$
6. You invest $1000 in an account that pays 5% interest compounded annually. How much will you have after 10 years?
   1. A. $1,500
   2. B. $1,628.89
   3. C. $1,795.86
   4. D. $2,000
7. The population of a town is growing at a rate of 3% per year. If the current population is 5000, what will the population be in 5 years?
   1. A. 5,796
   2. B. 5,800
   3. C. 5,650
   4. D. 5,000