Practical examples of exponential functions in business and science for Algebra 2

📚 Quick Study Guide

• 📈 Definition: An exponential function is a function of the form $f(x) = ab^x$, where $a$ is a non-zero constant, $b$ is the base and is a positive real number not equal to 1, and $x$ is a variable.
• 💰 Exponential Growth: When $b > 1$, the function represents exponential growth. This means the value of $f(x)$ increases rapidly as $x$ increases.
• 📉 Exponential Decay: When $0 < b < 1$, the function represents exponential decay. This means the value of $f(x)$ decreases rapidly as $x$ increases.
• 💸 Compound Interest: A common example is compound interest, modeled by the formula $A = P(1 + \frac{r}{n})^{nt}$, where:
  • $A$ = the future value of the investment/loan, including interest
  • $P$ = the principal investment amount (the initial deposit or loan amount)
  • $r$ = the annual interest rate (as a decimal)
  • $n$ = the number of times that interest is compounded per year
  • $t$ = the number of years the money is invested or borrowed for
• 🦠 Bacterial Growth: Exponential functions can model the growth of bacteria populations.
• ☢️ Radioactive Decay: Exponential decay models the decay of radioactive substances, using the formula $N(t) = N_0e^{-kt}$, where:
  • $N(t)$ = the amount of the substance remaining after time $t$
  • $N_0$ = the initial amount of the substance
  • $k$ = the decay constant
  • $t$ = time

🧪 Practice Quiz

1. Question 1: Which of the following functions represents exponential growth?
   1. A. $f(x) = 0.5^x$
   2. B. $f(x) = -2^x$
   3. C. $f(x) = 3^x$
   4. D. $f(x) = x^3$
2. Question 2: A bacteria population doubles every hour. If you start with 50 bacteria, which function models the population after $t$ hours?
   1. A. $P(t) = 50 + 2t$
   2. B. $P(t) = 50 \cdot t^2$
   3. C. $P(t) = 50 \cdot 2^t$
   4. D. $P(t) = 2 \cdot 50^t$
3. Question 3: The half-life of a radioactive substance is 10 years. If you start with 200 grams, how much will remain after 30 years? (Use the concept of exponential decay)
   1. A. 100 grams
   2. B. 50 grams
   3. C. 25 grams
   4. D. 12.5 grams
4. Question 4: Which formula represents compound interest, where A is the future value, P is the principal, r is the rate, n is the number of times compounded per year, and t is the time in years?
   1. A. $A = P(1 + rt)$
   2. B. $A = P(1 + r)^t$
   3. C. $A = P(1 + \frac{r}{n})^{nt}$
   4. D. $A = P(1 + r)^{nt}$
5. Question 5: If \$1000 is invested at an annual interest rate of 5% compounded annually, what will be the value of the investment after 5 years?
   1. A. \$1200.00
   2. B. \$1276.28
   3. C. \$1250.00
   4. D. \$1050.00
6. Question 6: A car depreciates at a rate of 15% per year. If the original price was \$25,000, what is its value after 3 years?
   1. A. \$15,343.75
   2. B. \$16,343.75
   3. C. \$17,343.75
   4. D. \$18,343.75
7. Question 7: Which of the following is NOT an example of an exponential function in real-world applications?
   1. A. Population growth
   2. B. Radioactive decay
   3. C. Linear depreciation
   4. D. Compound interest