Exponential change multiplies by the same factor each period, where linear change adds the same amount.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The model
y = a · bˣ, where a is the starting amount and b is the factor applied each period.
Reading the factor
A factor above 1 grows, below 1 decays. A 20% rise gives b = 1.2 and a 20% fall gives b = 0.8.
The factor is not the rate
Writing b = 0.2 for a 20% rise is the standard error. The factor includes the original amount.
The shape
Growth starts slowly and becomes very steep. Decay falls fast and then flattens, approaching but never reaching zero.
It always wins eventually
Any exponential growth overtakes any linear growth, given enough time. It may take a while.
Multiplying instead of adding
A linear function adds a constant each step; an exponential multiplies by a constant factor. That single difference produces radically different long-run behaviour.
The form and its parts
y = abˣ, with a the starting value and b the growth factor. Growth when b exceeds 1, decay when it lies between 0 and 1. A percentage change of r gives b = 1 + r.
Exponentials eventually win
Any exponential with base above 1 overtakes any linear function, however steep, and then leaves it far behind. The crossover may be late, but it always arrives.
Where it applies
Compound interest, population growth, radioactive decay and the early phase of an epidemic. The common feature is that the change depends on how much is already present.
Step 2: Try It Yourself
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Step 3: Watch an Example
One step at a time.
Watch Tomas Model a Population
A town of 5000 people grows 3% a year.
- Step 1
The starting amount is a = 5000.
Step 4: Your Turn
Practice makes it stick.
The Factor
Problem 1 of 2
A 25% annual rise. What is the growth factor?
The Decay
Problem 2 of 2
A 40% annual fall. What is the decay factor?
Grow and Shrink
1 of 8
A 10% rise. Growth factor?
2 of 8
A 50% fall. Decay factor?
3 of 8
y = 100(2)ˣ at x = 3. What is y?
4 of 8
y = 80(0.5)ˣ at x = 3. What is y?
5 of 8
y = 200(1.5)ˣ at x = 2. What is y?
6 of 8
Does an exponential eventually overtake any linear growth? 1 yes, 0 no.
7 of 8
Sort each situation by the type of change.
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8 of 8
y = 50(3)ˣ at x = 2. What is y?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A 30% annual rise. What is the growth factor?
Question 2 of 2
What distinguishes exponential from linear change?
What You Learned
- Exponential change multiplies by a fixed factor each period.
- The model is y = a · bˣ, with b above 1 for growth and below 1 for decay.
- The factor is 1 plus the rate, not the rate alone.