Mathematics • Year 9 • Unit 2 • Lesson 14

Doubling, Tripling, Spreading — Exponentials in Real Life

Use $y = a^x$ to model a doubling rumour, a tripling bacteria colony, savings with daily compounding, viral video views, and a school sport tournament bracket. Spot how fast repeated multiplication runs away from repeated addition.

Apply · Real-World Maths

1. Word problems

Each scenario is exponential. Build small tables, write the formula, and answer in full sentences with units. 3 marks each

1.1 — Doubling rumour. One student starts a rumour at 8 a.m. Every hour the number of students who know it doubles. Let $h$ = hours since 8 a.m. and $N$ = number who know.

(a) Write the formula for $N$ in terms of $h$.
(b) How many students know it at 8 a.m., 9 a.m., 10 a.m., 12 noon, and 2 p.m.?
(c) The school has 1024 students. After how many hours does EVERYONE know?

Stuck on (c)? $1024 = 2^{10}$, so $h = 10$ hours.

1.2 — Tripling bacteria. A bacterial culture starts with $4$ cells. Every hour, the number TRIPLES. Let $t$ = hours and $N$ = number of cells.

(a) Write the formula for $N$ in terms of $t$.
(b) Build a table for $t = 0, 1, 2, 3$.
(c) How many cells are there after $5$ hours?

Stuck? Start at $4$, multiply by $3$ each hour: $N = 4 \times 3^t$.

1.3 — Viral video views. A new music video starts with 100 views and the daily view count doubles each day for the first week. Let $d$ = days after release and $V$ = total views (the simple model: $V = 100 \times 2^d$).

(a) Find $V$ at $d = 0, 1, 2, 3, 7$.
(b) Is $V$ growing linearly or exponentially? Justify using the first-differences pattern from your table.
(c) Roughly when does $V$ first cross $10\,000$ views?

Stuck on (c)? $100 \times 2^7 = 12\,800$, $100 \times 2^6 = 6\,400$. So it first crosses $10\,000$ between day $6$ and day $7$.

1.4 — Tournament bracket. A school sport tournament starts with 32 teams. Each round, half the teams are knocked out (so the number remaining halves). Let $r$ = rounds played and $T$ = teams remaining.

(a) Build a table for $r = 0, 1, 2, 3, 4, 5$.
(b) Write a formula for $T$ in terms of $r$.
(c) Is this exponential? If so, is it growth or decay?

Stuck? Starting with 32 and halving each round: $T = 32 \times (1/2)^r = 32/2^r$. After $r = 5$ rounds, $T = 1$ (the winner).

1.5 — Linear savings vs exponential growth. Tom adds $\$50$ to a piggy bank each week, starting at $\$0$. Sara starts with $\$1$ and her balance DOUBLES every week (no idea how, magic). Let $w$ = weeks. For Tom $B_T = 50w$; for Sara $B_S = 2^w$.

(a) Build a table of $B_T$ and $B_S$ at $w = 0, 2, 4, 6, 8, 10, 12$.
(b) At which week does Sara's balance first exceed Tom's, and stay ahead?
(c) Explain in one sentence why this happens, using the words "linear" and "exponential".

Stuck? Compute both: at $w = 10$, Tom has $\$500$, Sara has $\$1024$. Sara has just edged ahead.

2. Explain your thinking

Use full sentences, no dot points. 4 marks

2.1 A classmate is asked for the $y$-intercept of $y = 2^x$ and says "$(0, 0)$, because the line goes through the origin like any other graph." In your own words, explain (i) why their answer is wrong, (ii) what $2^0$ actually equals and why, (iii) what the correct $y$-intercept is, and (iv) how this is a fundamental difference between $y = 2x$ (which DOES go through the origin) and $y = 2^x$ (which does NOT).

Stuck? Revisit lesson § "Common Pitfalls" — the "$2^0 = 0$ error" row is exactly this misconception.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Doubling rumour

(a) $\mathbf{N = 2^h}$.
(b) $h = 0, 1, 2, 4, 6$ gives $N = 1, 2, 4, 16, 64$ students.
(c) $1024 = 2^{10}$, so $h = \mathbf{10}$ hours after 8 a.m. (6 p.m.).

1.2 — Tripling bacteria

(a) $\mathbf{N = 4 \times 3^t}$.
(b) $t = 0, 1, 2, 3$ gives $N = 4, 12, 36, 108$ cells.
(c) $N(5) = 4 \times 3^5 = 4 \times 243 = \mathbf{972}$ cells.

1.3 — Viral views

(a) $V(0) = 100$, $V(1) = 200$, $V(2) = 400$, $V(3) = 800$, $V(7) = 100 \times 128 = 12\,800$.
(b) Exponential. First differences $100, 200, 400, \ldots$ are not constant (they double each step). For linear they would all be the same.
(c) $V(6) = 100 \times 64 = 6\,400 < 10\,000$. $V(7) = 12\,800 > 10\,000$. So $V$ first crosses $10\,000$ between day 6 and day 7.

1.4 — Tournament bracket

(a) $r = 0, 1, 2, 3, 4, 5$ gives $T = 32, 16, 8, 4, 2, 1$.
(b) $\mathbf{T = 32 \times \left(\tfrac{1}{2}\right)^r}$ (equivalently $T = 32/2^r$).
(c) Exponential decay — the base is $\tfrac{1}{2}$ ($0 < a < 1$), so $T$ shrinks by a constant ratio each round.

1.5 — Tom vs Sara

(a) Table:
$w$: 0, 2, 4, 6, 8, 10, 12
$B_T$ (linear): 0, 100, 200, 300, 400, 500, 600
$B_S$ (exponential): 1, 4, 16, 64, 256, 1024, 4096
(b) Sara is BEHIND Tom for many weeks (e.g. at $w = 8$ Sara has $\$256$ vs Tom's $\$400$). Sara first exceeds Tom at $w = 10$ ($\$1024$ vs $\$500$) and stays ahead from there.
(c) Tom's balance grows linearly (constant add of $\$50$/week), but Sara's grows exponentially (doubling each week) — eventually repeated multiplication always overtakes repeated addition.

2.1 — Explain your thinking (sample response)

My classmate is wrong because not every graph passes through the origin. The $y$-intercept is the value of $y$ when $x = 0$, and for $y = 2^x$ that means $y = 2^0$. The zero-exponent rule says $a^0 = 1$ for ANY positive base $a$ — that's just how exponents are defined. So $2^0 = 1$, not $0$, and the correct $y$-intercept is $\mathbf{(0, 1)}$. This is a key difference between $y = 2x$ (which is linear with $y$-intercept $(0, 0)$ — $2 \times 0 = 0$) and $y = 2^x$ (which is exponential with $y$-intercept $(0, 1)$ — $2^0 = 1$). Same numbers, totally different roles: in $2x$ the $x$ is multiplying, in $2^x$ the $x$ is the exponent.

Marking: 1 mark for naming the zero-exponent rule; 1 mark for the correct intercept $(0, 1)$; 1 mark for distinguishing $2x$ from $2^x$; 1 mark for clear writing.