Exponential growth and decay: multiplying by a constant factor

Discover

Two ponds each start with 5 m² of algae. In pond A, the area grows by 5 m² each week. In pond B, the area doubles each week.

After 1 week both ponds have 10 m². Which pond has more algae after 4 weeks? Guess first.

Pond A
Pond B
They have the same

Then check with a small table for weeks 0 to 4.

Key idea

When a quantity is multiplied by the same factor $b$ at each step, it changes exponentially.

Start

a

, factor

b

each step: after

n

steps the amount is

a \cdot b^n

If $b > 1$ , the quantity grows (exponential growth).
If $0 < b < 1$ , it shrinks (exponential decay).
Losing $p\%$ each step keeps $(100 - p)\%$ , so the factor is $1 - \frac{p}{100}$ . Gaining $p\%$ gives the factor $1 + \frac{p}{100}$ .

Worked example

A photo is 1000 pixels wide. Each time it is resized to 80% of its width. How wide is it after 3 resizes?

W = 1000 \cdot 0.8 \cdot 0.8 \cdot 0.8

Keeping 80% means multiplying by 0.8 each time.

W = 1000 \cdot 0.8^3

Three equal factors make a power.

W = 512

pixels

0.8³ = 0.512.

Your turn, guided

Your turn: a plant is 3 cm tall, and its height triples every week. How tall is it after 4 weeks? Fill in the missing line.

h = 3 \cdot 3^4

Start at 3 and multiply by 3 four times.

h = 243

cm

3⁴ = 81, and 3 · 81 = 243.

Practice ★

The amount of algae starts at 5 m² and is multiplied by 4 each week. Complete the table.
The amount of algae starts at 2 m² and is multiplied by 2 each week. Complete the table.

A culture starts with 7 grams of algae. Each week, the amount is multiplied by 3. How many grams are there after 3 weeks?
A culture starts with 5 grams of algae. Each week, the amount is multiplied by 2. How many grams are there after 5 weeks?
A culture starts with 3 grams of algae. Each week, the amount is multiplied by 3. How many grams are there after 5 weeks?
A culture starts with 6 grams of algae. Each week, the amount is multiplied by 5. How many grams are there after 5 weeks?
A culture starts with 5 grams of algae. Each week, the amount is multiplied by 2. How many grams are there after 3 weeks?

Watch out!

Practice ★★

A photo is 8000 pixels wide. Each time it is resized, its width loses 40%. Write the width $W(t)$ after $t$ resizes.
A photo is 2000 pixels wide. Each time it is resized, its width loses 30%. Write the width $W(t)$ after $t$ resizes.
A photo is 2000 pixels wide. Each time it is resized, its width loses 25%. Write the width $W(t)$ after $t$ resizes.
A photo is 8000 pixels wide. Each time it is resized, its width loses 10%. Write the width $W(t)$ after $t$ resizes.
A photo is 4000 pixels wide. Each time it is resized, its width loses 30%. Write the width $W(t)$ after $t$ resizes.

Exit ticket

Three questions to check you have got it.

A photo is 8000 pixels wide. Each time it is resized, its width loses 40%. Write the width $W(t)$ after $t$ resizes.
A photo is 2000 pixels wide. Each time it is resized, its width loses 30%. Write the width $W(t)$ after $t$ resizes.
A photo is 2000 pixels wide. Each time it is resized, its width loses 25%. Write the width $W(t)$ after $t$ resizes.

Summary card

Exponential change: the same factor at every step

Start $a$ , factor $b$ per step: after $n$ steps, $a \cdot b^n$ .
Gain $p\%$ : factor $1 + \frac{p}{100}$ . Lose $p\%$ : factor $1 - \frac{p}{100}$ .

Watch out: adding the same amount each step is linear, not exponential.

About this lesson