In March 2020, COVID-19 cases worldwide grew from hundreds to thousands to tens of thousands in weeks. This was not linear growth — it was exponential. Understanding exponential functions is not just a mathematical skill; it is essential for interpreting pandemics, compound interest, population dynamics, and every technology that improves by doubling.
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A bacteria colony doubles every hour. Starting with 100 bacteria, after 10 hours the population will be:
Make a prediction before reading on. Most people's intuition is linear — but exponential growth behaves very differently.
Type your initial prediction below — you will revisit this at the end of the lesson.
Write your initial prediction in your book. You will revisit it at the end of the lesson.
Core Content
An exponential function has the form:
$$y = a^x$$
where $a$ is a positive constant called the base, and $a \neq 1$. The variable $x$ appears in the exponent, which makes this fundamentally different from power functions like $y = x^2$ where the variable is in the base.
Key restriction: The base must be positive ($a > 0$). If $a$ were negative, say $a = -2$, then $(-2)^{1/2}$ would involve the square root of a negative number, which is not a real number. To keep the function defined for all real $x$, we require $a > 0$.
The two most important families are:
Every exponential function $y = a^x$ (with $a > 0$, $a \neq 1$) shares three universal features:
When $x = 0$:
$$y = a^0 = 1$$
Every exponential function passes through $(0, 1)$, regardless of the base. This is a powerful identification tool: if a curve passes through $(0, 1)$ and is always positive, it may be exponential.
As $x \to -\infty$ (for growth, $a > 1$) or $x \to +\infty$ (for decay, $0 < a < 1$), the function approaches zero:
$$y \to 0$$
The line $y = 0$ (the $x$-axis) is a horizontal asymptote. The function gets arbitrarily close to zero but never touches or crosses it. This is why exponential decay models (radioactive half-life, drug elimination) never quite reach zero — they asymptote toward it.
Show four exponential curves on one set of axes with $x$ from -3 to 3 and $y$ from 0 to 8. Curve A (blue): $y = 2^x$ passing through (-1, 0.5), (0, 1), (1, 2), (2, 4). Curve B (green): $y = 3^x$ passing through (-1, 0.33), (0, 1), (1, 3), (2, 9) — steeper than A. Curve C (orange): $y = (\frac{1}{2})^x = 2^{-x}$ passing through (-2, 4), (-1, 2), (0, 1), (1, 0.5), (2, 0.25) — decreasing. Curve D (red): $y = (\frac{1}{3})^x$ passing through (-2, 9), (-1, 3), (0, 1), (1, 0.33) — decreasing faster than C. All curves approach $y = 0$ as $x \to -\infty$ (for A, B) or $x \to +\infty$ (for C, D). Label the horizontal asymptote $y = 0$ as a dashed line. Mark the common $y$-intercept $(0, 1)$ with a dot.
The function $f(x) = 3 \cdot 2^x$ models the population of bacteria (in thousands) $x$ hours after observation begins.
(a) The initial population.
(b) The population after 4 hours.
(c) The value of $x$ when the population reaches 48,000.
(a) Initial = f(0) = 3 · 2⁰ = 3 · 1 = 3 thousand
(b) f(4) = 3 · 2⁴ = 3 · 16 = 48 thousand
(c) 3 · 2ˣ = 48
2ˣ = 16
2ˣ = 2⁴
x = 4 hours
(a) 3,000 bacteria
(b) 48,000 bacteria
(c) 4 hours
The value of a painting is modelled by $V(t) = 5000 \cdot (1.08)^t$ where $t$ is years since purchase.
Answer:
(1) $V(0) = 5000$ dollars
(2) $V(10) = 5000 \cdot (1.08)^{10} \approx 10,794.62$, so $\$10,795$
(3) $5000 \cdot (1.08)^t > 10000 \Rightarrow (1.08)^t > 2 \Rightarrow t > \frac{\ln 2}{\ln 1.08} \approx 9.01$, so 10 years.
While any positive base $a \neq 1$ defines a valid exponential function, three bases dominate mathematics, science, and finance:
$y = 2^x$ appears wherever quantities double or halve: cell division, Moore's Law (transistor counts), binary computing, and half-life calculations. Each increment of $x$ by 1 doubles the output.
$y = 10^x$ is the foundation of scientific notation, the Richter scale, decibels, and pH. Each increment of $x$ by 1 multiplies the output by 10. This is why "orders of magnitude" means powers of 10.
$e \approx 2.71828$ is the most important base in calculus. The function $y = e^x$ has the unique property that its rate of change at any point equals its value at that point — a property that makes it indispensable for modelling continuous growth, decay, and every physical process described by differential equations. We dedicate Lesson 3 to understanding why $e$ is special.
$y = a^x$ where $a > 0$, $a \neq 1$
Domain: $x \in \mathbb{R}$
Range: $y > 0$
Always $(0, 1)$ since $a^0 = 1$
Horizontal asymptote: $y = 0$
Growth: $a > 1$; Decay: $0 < a < 1$
For each function, calculate the missing values and state whether the function shows growth or decay.
A student claims that $y = (-2)^x$ is an exponential function because it has the form $a^x$. Another student says $y = 1^x$ is also exponential.
Look back at what you wrote in the Think First section. The bacteria colony doubles every hour starting from 100. After 10 hours, the population is $100 \times 2^{10} = 100 \times 1024 = 102,400$ bacteria.
5 random questions from a replayable lesson bank — feedback shown immediately
Short Answer
8. The population of a town is modelled by $P(t) = 2000 \cdot (1.05)^t$ where $t$ is years since 2020. (a) Find the population in 2020. (b) Find the population in 2030 (to the nearest person). (c) In what year will the population first exceed 3,500? Show your working. 3 MARKS
Type your answer below:
Answer in your workbook.
9. Consider the functions $f(x) = 3^x$ and $g(x) = (\frac{1}{3})^x$. (a) Complete a table of values for $x = -2, -1, 0, 1, 2$ for both functions. (b) Sketch both curves on the same set of axes, labelling key features. (c) Explain the relationship between $f(x)$ and $g(x)$ algebraically. 3 MARKS
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Answer in your workbook.
10. During the early phase of the COVID-19 pandemic, epidemiologists modelled case growth as exponential. Explain why early intervention (such as social distancing) is far more effective at reducing total cases when implemented in week 1 rather than week 4 of an outbreak. Use the concept of exponential growth and specific numerical examples to support your answer. 3 MARKS
Type your answer below:
Answer in your workbook.
1. $f(-2) = 2^{-2} = \frac{1}{4}$; $f(0) = 2^0 = 1$; $f(3) = 2^3 = 8$. Growth (base 2 > 1).
2. $g(-1) = (\frac{1}{3})^{-1} = 3$; $g(0) = (\frac{1}{3})^0 = 1$; $g(2) = (\frac{1}{3})^2 = \frac{1}{9}$. Decay (base between 0 and 1).
3. $h(0) = 5 \cdot 2^0 = 5$; $h(2) = 5 \cdot 2^2 = 20$; $5 \cdot 2^x = 40 \Rightarrow 2^x = 8 = 2^3 \Rightarrow x = 3$.
1. $y = (-2)^x$ is not exponential because the base must be positive. For $x = \frac{1}{2}$, $(-2)^{1/2} = \sqrt{-2}$ which is not a real number. The function is undefined for many real $x$ values.
2. $y = 1^x = 1$ for all $x$. This is a constant function, not exponential. The definition requires $a \neq 1$.
3. An exponential (i) passes through $(0, 1)$, (ii) is always above the $x$-axis, (iii) has $y = 0$ as a horizontal asymptote, (iv) is either strictly increasing (growth) or strictly decreasing (decay).
Q8 (3 marks): (a) $P(0) = 2000 \cdot (1.05)^0 = 2000$ people [1]. (b) $P(10) = 2000 \cdot (1.05)^{10} \approx 2000 \cdot 1.6289 \approx 3257.79$, so 3258 people [1]. (c) $2000 \cdot (1.05)^t > 3500 \Rightarrow (1.05)^t > 1.75 \Rightarrow t > \frac{\ln 1.75}{\ln 1.05} \approx 11.34$. So $t = 12$ years, meaning 2032 [1].
Q9 (3 marks): (a) $f(x) = 3^x$: $(-2, \frac{1}{9}), (-1, \frac{1}{3}), (0, 1), (1, 3), (2, 9)$. $g(x) = (\frac{1}{3})^x$: $(-2, 9), (-1, 3), (0, 1), (1, \frac{1}{3}), (2, \frac{1}{9})$ [1]. (b) Both pass through $(0, 1)$ with asymptote $y = 0$. $f(x)$ increases; $g(x)$ decreases. $f$ is steeper than $g$ is shallow [1]. (c) $g(x) = (\frac{1}{3})^x = 3^{-x} = f(-x)$. So $g$ is the reflection of $f$ in the $y$-axis [1].
Q10 (3 marks): With exponential growth, each infected person infects multiple others, creating a multiplicative cascade [1]. If cases double weekly, week 1 has 100 cases, week 4 has 800 cases. Intervention in week 1 prevents 700 cases directly plus all downstream infections from those 700. By week 4, the same proportional reduction removes far more absolute cases because the base is larger [1]. Exponential growth means delays are amplified — a 3-week delay at doubling rate produces $2^3 = 8$ times more cases, making late intervention exponentially less effective [1].
Climb platforms using your knowledge of exponential functions, domain, range, and growth vs decay. Pool: lesson 1.
Tick when you've finished all activities and checked your answers.