Exponential Functions – Growth and Decay
An exponential function has the form y = a · bx, where the variable x sits in the exponent rather than being multiplied by a number. This single change makes exponential functions behave completely differently from anything in ordinary algebra: instead of growing (or shrinking) by a fixed amount each step, they grow or shrink by a fixed percentage each step – which is exactly why compound interest, population growth, and epidemics all eventually explode far faster than most people expect.
The Swiss mathematician Jacob Bernoulli discovered the mathematical constant e (≈ 2.71828) around 1683 while studying compound interest – specifically, what happens to your money if interest is compounded not once a year but continuously. Leonhard Euler later named the constant e in his honour and showed how central it is to exponential growth in general. Long before the formal mathematics existed, the ancient legend of the inventor of chess asking for one grain of rice on the first square, doubling on every square after (1, 2, 4, 8, 16…), was already illustrating exponential growth: by the 64th square, the rice required exceeds all the rice ever produced on Earth.
What Is an Exponential Function?
y = a · bx, where a is the starting value, b is the growth factor (b > 1 means growth, 0 < b < 1 means decay), and x is the number of time periods.
y = 100 × 2x, with x = 5: y = 100 × 25 = 100 × 32 = 3,200 bacteria.
Exponential Growth: Compound Interest
A = P(1 + r)t, where P is the starting amount, r is the growth rate as a decimal, and t is the number of time periods.
A = 1000(1.08)10 ≈ $2,158.92 – more than double, even though the rate is only 8% per year.
Exponential Decay
When 0 < b < 1, the function shrinks toward zero but never quite reaches it – the x-axis (y = 0) is a horizontal asymptote. Radioactive decay, cooling coffee, and depreciating car values all follow this pattern.
15 years is 3 half-lives, so the amount halves 3 times: 80 → 40 → 20 → 10 g.
Real-Life Applications
- Finance: Compound interest, loan balances, and investment growth.
- Biology: Bacteria and population growth, and the spread of epidemics.
- Physics and Chemistry: Radioactive decay and carbon dating.
- Technology: Processing power and data storage have historically grown exponentially (Moore's Law).
- Everyday life: A car's value depreciating, or a rumour spreading through a school.
Key Takeaways
- Exponential functions have the form y = a · bx, with the variable in the exponent.
- b > 1 means growth; 0 < b < 1 means decay, always approaching y = 0 but never reaching it.
- Compound interest A = P(1 + r)t is exponential growth in action.
- The next page, Logarithms, covers the inverse operation – how to work backwards from an exponential result to find x.