Lesson 57: Finding Exponential Growth and Decay

New Concept

Exponential growth is modeled by the function $f(x) = ab^x$, where $a > 0$ and $b > 1$.

What’s next

Next, you’ll apply this model to calculate compound interest, analyze radioactive decay, and derive exponential equations from real-world data.

Exponential growth models a quantity that always increases by the same percent over time. It is represented by the function $f(x) = ab^x$, where 'a' is the initial amount, 'b' is the growth factor ($b > 1$), and 'x' is the number of time intervals. This is perfect for situations like population or investment growth!

A town's population starts at 5,000 and grows by 3% per year. The model is $y = 5000(1.03)^t$. After 10 years, the population is $5000(1.03)^{10} \approx 6719$.
A scientist finds 100 bacteria that double every hour. The model is $y = 100(2)^h$. After 8 hours, there are $100(2)^8 = 25600$ bacteria.

Think of a super-powered piggy bank! Your initial cash 'a' doesn't just grow, it multiplies by the growth factor 'b' over and over. Your savings don't just increase, they skyrocket!

Exponential decay models a quantity that always decreases by the same percent over a period of time. It uses the function $f(x) = ab^x$, where 'a' is the starting amount and the decay factor 'b' is between 0 and 1 ($0 < b < 1$). This can also be written as $f(x) = ab^{-x}$ where $b > 1$.

A car worth 25000 dollars depreciates by 15% each year. The model is $y = 25000(0.85)^t$. After 5 years, its value is $25000(0.85)^5 \approx 11092.63$ dollars.
A pendulum's swing starts at 2 meters. Each swing is 90% of the previous one. The model is $y = 2(0.9)^s$. The 4th swing will be $2(0.9)^4 = 1.3122$ meters.

Imagine your phone battery draining. It starts full ('a') but loses a fraction ('b') of its remaining power each hour. The drop is steepest at the start, then slows as it gets closer to zero.

When interest is compounded continuously, it means the number of compounding periods is infinite. This special case of exponential growth uses the mathematical constant 'e' (approximately 2.718). The formula to find the value of an account, A, is given by $A = Pe^{rt}$, where P is the principal, r is the annual interest rate, and t is the time in years.

You deposit 1000 dollars at a 4% annual rate, compounded continuously for 3 years. Using $A = Pe^{rt}$, you get $A = 1000e^{(0.04)(3)}$, which is approximately 1127.50 dollars.
An account is worth 5000 dollars after 6 years with 3% interest compounded continuously. The initial principal was $5000 = Pe^{(0.03)(6)}$, so $P \approx 4176.35$ dollars.

This is the ultimate interest plan! Instead of waiting, the bank adds a tiny bit of interest every single instant. This nonstop compounding, powered by the magic number 'e', helps your money grow as fast as possible.