Mathematics › 🌆 Grade 9 › Zombie Outbreak: The Knee of Exponential Growth

Zombie Outbreak: The Knee of Exponential Growth

One infection doubling every hour looks harmless for a while — until the curve hits its knee and the numbers explode.

Grade 9Algebra 1

Zombie Outbreak: The Knee of Exponential Growth — illustration

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The big idea: Exponential growth multiplies the same quantity by a fixed factor every time period, instead of adding a fixed amount. That single difference — multiplying instead of adding — is why exponential quantities look almost flat at first, then bend sharply upward at a point often called the “knee” of the curve, even though the growth rule itself never changed.

🎯 By the end, you'll be able to

Write an exponential growth function in the form y = a·b^t
Distinguish exponential growth (multiplying each period) from linear growth (adding each period)
Identify the initial value a and the growth factor b in a model
Explain why exponential growth appears slow at first and then accelerates sharply

📎 You should already know

Exponents with whole-number powers
Function notation (f(x))

One becomes two becomes four…

Imagine a single infected zombie appears, and the outbreak doubles every hour: 1, then 2, then 4, then 8. For the first few hours the numbers feel manageable. But doubling is relentless — by hour 10 that lone zombie has become over a thousand, and by hour 20, over a million. Nothing about the rule changed; multiplying by 2 again and again simply compounds.

🔑 Exponential growth multiplies, it doesn't add

In exponential growth, a quantity is multiplied by the same growth factor every time period. This is fundamentally different from linear growth, where the same fixed amount is added every period.

\[ y = a \cdot b^{t} \]

a is the starting amount (at t = 0), b is the growth factor per time period, and t is the number of periods elapsed. For growth, b > 1.

🎮 Zombie Outbreak LIVE

Grow an epidemic exponentially and find the 'knee' where slow becomes explosive.

Why the curve seems to have a 'knee'

Doubling from 1 to 2 is an increase of 1. Doubling from 512 to 1024 is an increase of 512 — the exact same growth factor, but a vastly bigger jump in absolute numbers. On a graph, the early values sit so close to the axis that the curve looks almost flat, then suddenly appears to bend sharply upward. That bend is often called the knee of the curve — but nothing changed at that point except how big the numbers had already become.

📝 Worked example: A single zombie doubles every hour: y = 1 · 2^t. How many zombies are there after 10 hours?

Substitute t = 10 into y = 1 · 2^t.
Compute 2^10 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024.
Multiply by the starting amount, 1.

✓ After 10 hours there are <strong>1024 zombies</strong> — over a thousand, from a single starting case.

📝 Worked example: A town of 100 people grows at 5% per year: y = 100(1.05)^t. About how large is the population after 10 years?

The growth factor is 1.05 (100% plus the 5% increase, written as a decimal).
Compute (1.05)^10 ≈ 1.629.
Multiply: y ≈ 100 × 1.629.

✓ The population is about <strong>163 people</strong> after 10 years.

✨ The growth factor tells you the percent change

A growth factor of 1.05 means the quantity keeps 100% of itself and adds another 5% — a 5% increase per period. A growth factor of 2 means the quantity doubles — a 100% increase. The growth factor is always 1 plus the rate, written as a decimal.

⚠️ Don't extrapolate exponential growth like a straight line

It is tempting to think “it grew by 1 zombie in the first hour, so it grows by 1 every hour.” That is linear thinking applied to an exponential process, and it badly underestimates the outcome. Exponential growth means the rate of change itself keeps increasing.

Exponential growth is everywhere

Compound interest, viral spread, and population growth under unlimited resources all follow this same multiply-each-period pattern. Recognizing y = a · b^t is the first step to modeling any of them.

Check your understanding

1. In the long run, which grows faster: exponential growth or linear growth?

Exponential growth compounds — it multiplies by the same factor every period — so it eventually outpaces any linear (constant-addition) growth, no matter the starting values.

2. A colony starts with 3 organisms and doubles every hour. How many are there after 4 hours?

y = 3 · 2⁴ = 3 × 16 = 48.

3. In y = 50(1.2)^t, what does the growth factor 1.2 represent?

A growth factor of 1.2 equals 100% + 20%, meaning the quantity increases by 20% each period.

4. Which best describes exponential growth compared to linear growth?

The defining feature of exponential growth is multiplying by a constant factor each period, rather than adding a constant amount.

5. Does an exponential growth curve ever stop accelerating?

In true exponential growth (b > 1), the amount added each period keeps getting bigger, so the growth never levels off into a constant rate.

✅ Key takeaways

Exponential growth multiplies a quantity by a fixed factor each period; linear growth adds a fixed amount each period.
y = a·b^t models exponential growth, with a as the starting value and b as the growth factor (b > 1).
Growth that looks slow at first and then bends sharply upward hasn't changed its rule — the absolute jumps just get bigger as the quantity grows.
A growth factor of 1 + r represents an r% increase per period; for example 1.05 is a 5% increase.
Never extrapolate exponential growth using a constant, linear amount per period — the rate of increase itself keeps growing.