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3.13a · Exponential Functions · Sub-skill
Model growth and decay with $y = ab^x$ and compare it with linear change.
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A quantity grows exponentially when it is multiplied by the same factor in each equal time step — e.g. bacteria that double every hour: 1, 2, 4, 8, 16, … Equal steps give equal ratios, unlike linear growth where equal steps add equal amounts.
An exponential function is $y = ab^x$, where $a$ is the starting value (the $y$-intercept, since $b^0 = 1$) and $b$ is the multiplier for each step. For example $N = 500 \times 2^t$ starts at 500 and doubles each hour.
If $b > 1$ the function shows growth; if $0 < b < 1$ it shows decay. A percentage increase of $r\%$ has multiplier $1 + \frac{r}{100}$ and a decrease has multiplier $1 - \frac{r}{100}$. A quantity that halves every step has $b = 0.5$.
Linear growth adds the same amount each step; exponential growth multiplies. Exponential growth starts slowly but eventually overtakes any linear or quadratic growth. Real exponential models only hold for a time — populations run out of resources.
500 bacteria double every hour. Write a formula for the number $N$ after $t$ hours and find $N$ after 5 hours.
$N = 500 \times 2^t$.
After 5 hours: $N = 500 \times 2^5 = 500 \times 32 = $ 16 000.
Test yourself
The population of a town is 8000 and it grows by 3% each year.
(a) Write down a formula for the population $P$ after $t$ years.
(b) Find the population after 10 years, to the nearest whole number.
(c) A second town grows by a constant 240 people a year from 8000. Which town is bigger after 10 years, and by how much? [5 marks]
(a) $P = 8000 \times 1.03^t$. (1 mark)
(b) $8000 \times 1.03^{10} = 8000 \times 1.3439 = $ 10 751. (2 marks)
(c) Second town: $8000 + 10 \times 240 = 10\,400$. The first town is bigger by $10\,751 - 10\,400 = $ 351. (2 marks)
Practice
Five short questions. Work through them, then reveal the mark scheme to check.