Exponential functions grow or decay by a constant multiplicative factor. Like bacteria doubling every hour or radioactive material halving over time, these functions model processes where the rate of change is proportional to the current amount.
Today's hook, A sheet of paper is 0.1 mm thick. You fold it in half 10 times. How tall is the stack? Most people guess under 10 cm. The real answer is $2^{10} = 1024$ layers, about 10 cm. Fold 42 times and it reaches the Moon. That explosive growth is exactly what this lesson is about.
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Orient and recall
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
A piece of paper is folded in half 10 times. Roughly how many layers thick is it? Pick a number first, then explain why you chose it, no formula needed yet.
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The two moves
Work through the core explanation before applying it.
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The two moves
+5 XP to read
Every exponential question in this module comes down to one decision: identify the base, then decide if it causes growth or decay. Everything else follows automatically.
For $y = a^x$: if $a > 1$ the function grows as $x$ increases; if $0 < a < 1$ the function decays. Both always pass through $(0, 1)$ since $a^0 = 1$ for any valid base.
$$y = a^x \quad a > 0,\ a \neq 1$$
Base > 1 (growth)
$y$ increases without bound as $x \to +\infty$ and approaches 0 as $x \to -\infty$.
0 < Base < 1 (decay)
$y$ approaches 0 as $x \to +\infty$ and grows without bound as $x \to -\infty$.
Negative exponents
$a^{-n} = \dfrac{1}{a^n}$, a negative exponent creates a reciprocal, not a negative value.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
The definition of an exponential function $y = a^x$
The conditions on the base: $a > 0$, $a \neq 1$
That $a^0 = 1$ for all valid bases
Understand
Concepts
Why $a > 1$ produces growth and $0 < a < 1$ produces decay
The domain (all real $x$) and range ($y > 0$) of $y = a^x$
How exponential growth differs from polynomial growth
Can do
Skills
Evaluate $a^x$ for positive, negative and fractional values of $x$
Classify a given base as growth or decay
Apply exponential models to real-world problems
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Key terms
Exponential functionA function of the form $y = a^x$ where $a > 0$ and $a \neq 1$.
BaseThe constant $a$ in $y = a^x$; determines whether the function shows growth or decay.
Exponential growthOccurs when $a > 1$: as $x$ increases, $y$ increases without bound.
Exponential decayOccurs when $0 < a < 1$: as $x$ increases, $y$ approaches zero.
Horizontal asymptoteThe line $y = 0$ that the basic exponential curve approaches but never crosses.
Domain / RangeFor $y = a^x$: domain is all real $x$; range is $y > 0$.
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What is an exponential function?
Work through the core explanation before applying it.
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What is an exponential function?
core concept
An exponential function has the variable in the exponent rather than the base. This creates dramatically different behaviour from polynomial functions. For $y = a^x$ with $a > 1$, each unit increase in $x$ multiplies $y$ by $a$, that is growth by a constant factor, not a constant amount.
$$y = a^x \quad \text{where } a > 0,\ a \neq 1$$
The function always passes through $(0, 1)$ because $a^0 = 1$ for any valid base. The $x$-axis ($y = 0$) is a horizontal asymptote, the graph gets infinitely close but never touches it. Exponential functions model population growth, compound interest, radioactive decay, and Newton's Law of Cooling.
Why the restriction $a \neq 1$? If $a = 1$, then $1^x = 1$ for every value of $x$, that is just the constant function $y = 1$, not interesting or exponential. If $a \le 0$, then $a^{1/2}$ may not be real. Both restrictions keep the function well-defined for all real $x$.
An exponential function has the variable in the exponent: $y = a^x$ where $a > 0$, $a \neq 1$; Always passes through $(0, 1)$ since $a^0 = 1$
Pause, copy the exponential function definition ($y = a^x$, $a > 0$, $a \neq 1$) and the universal fixed point ($a^0 = 1$, so all pass through $(0,1)$) into your book.
Did you get this? True or false: the function $y = (-2)^x$ is a valid exponential function because $|-2| > 1$.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · EVALUATING
Evaluate $f(x) = 2^x$ at $x = -2,\ 0,\ 3$.
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$f(-2) = 2^{-2} = \dfrac{1}{2^2} = \dfrac{1}{4}$
Negative exponent means reciprocal: $a^{-n} = \frac{1}{a^n}$.
Determine whether $y = \left(\dfrac{1}{3}\right)^x$ shows growth or decay, and find the $y$-intercept.
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Base $a = \dfrac{1}{3}$
Identify the base of the exponential function.
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Since $0 < \dfrac{1}{3} < 1$, this is exponential decay.
Bases strictly between 0 and 1 produce decay.
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At $x = 0$: $y = \left(\dfrac{1}{3}\right)^0 = 1$. The $y$-intercept is $(0, 1)$.
$a^0 = 1$ for any valid base, the $y$-intercept is always $(0, 1)$.
PROBLEM 3 · REAL-WORLD MODEL
A population of bacteria doubles every hour, modelled by $P = 100 \times 2^t$. Find the population after 5 hours.
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$P = 100 \times 2^t$ where $t$ is in hours
The initial population is 100 and the growth factor is 2 per hour.
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At $t = 5$: $P = 100 \times 2^5 = 100 \times 32$
Substitute $t = 5$ into the formula.
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$P = 3200$ bacteria
$2^5 = 32$; $100 \times 32 = 3200$.
Quick check: What is $f(-3)$ for $f(x) = 2^x$?
Common errors · the 3 traps that cost marks
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Dodge the traps, then apply
Meet the mistakes that cost marks, then do it yourself.
Trap 01
Confusing exponential growth with polynomial growth
$2^x$ grows much faster than $x^2$. For large $x$, exponential functions eventually outpace any polynomial. Students sometimes think $x^2$ grows faster because the notation looks similar.
Trap 02
Forgetting that $a^{-n} = \frac{1}{a^n}$, not $-a^n$
Negative exponents create reciprocals, not negative answers. $2^{-3} = \frac{1}{8}$, not $-8$. This is a common arithmetic error when evaluating exponential functions.
Trap 03
Including $a = 1$ or $a \le 0$ as valid bases
$y = 1^x$ is just the constant $y = 1$, not exponential. For $a \le 0$, the function is not defined for all real $x$ (e.g. $(-2)^{1/2}$ is not real).
Fill in the blank: For the exponential function $y = a^x$ to be valid, we require $a$ ___ 0 and $a$ ___ 1.
Quick-fire practice · 5 problems
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Drill it, then lock it in
Run the quick drill and copy the summary into your book.
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Evaluate $3^4$ and $3^{-2}$.
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Does $y = 5^x$ show growth or decay? Explain.
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Find the $y$-intercept of $y = \left(\dfrac{2}{3}\right)^x$.
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Evaluate $f(x) = 10^x$ at $x = -1,\ 0,\ 2$.
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A substance decays by half each year. Starting at 800 g, how much remains after 3 years? Write the model and evaluate it.
Odd one out, Three of these represent valid exponential functions. Which one does NOT?
Two truths, one lie, Three statements about $y = a^x$ where $a > 1$. Which one is the lie?
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Revisit your thinking
Earlier you guessed how thick a paper would be after 10 folds. The answer is $2^{10} = 1024$ layers, about 10 cm. That doubling pattern is exactly what an exponential function with base 2 captures. Whether the base is greater than 1 (growth) or between 0 and 1 (decay), the same structure $y = a^x$ governs the behaviour.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
Write full responses, then check them against the model answers.
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Short answer
ApplyBand 43 marks
Q1. Evaluate $f(x) = \left(\dfrac{1}{2}\right)^x$ at $x = -2,\ 0,\ 4$. Show your working. (3 marks)
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ApplyBand 43 marks
Q2. The value of a car depreciates by 15% per year, modelled by $V = 30\,000 \times (0.85)^t$. Find the value after 2 years and after 4 years, to the nearest dollar. (3 marks)
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AnalyseBand 53 marks
Q3. For the function $f(x) = a^x$ where $a > 1$, explain why $f(x) > 0$ for all real $x$, and why $f(x) \to 0$ as $x \to -\infty$. (3 marks)
Q3 (3 marks): Any positive number raised to any real power is positive, so $a^x > 0$ for all $x$ [1]. As $x \to -\infty$, write $a^x = \frac{1}{a^{-x}}$ where $-x \to +\infty$ [1]. Since $a > 1$, $a^{-x} \to \infty$, so $\frac{1}{a^{-x}} \to 0$ [1].
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Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
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Take the full module quiz
quiz
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.
