The big idea: Use an exponential model y = a·bˣ when the quantity multiplies by the same factor each equal period. If b > 1, it grows. If 0 < b < 1, it decays.
| Parameter | Role | Example |
|---|---|---|
| a | Starting value (y when x = 0) | a = 500 → starts at 500 bacteria |
| b > 1 | Growth factor (multiplied each period) | b = 1.2 → 20% increase per period |
| 0 < b < 1 | Decay factor (multiplied each period) | b = 0.8 → 20% decrease per period |
Exponential growth (b > 1)
- Curve rises faster and faster
- Never negative (always above y = 0)
- Example: b = 1.05 means 5% increase per period
Exponential decay (0 < b < 1)
- Curve falls towards y = 0 but never reaches it
- Horizontal asymptote at y = 0
- Example: b = 0.9 means 10% decrease per period
Growth or decay?: b > 1 → growth (curve rises). 0 < b < 1 → decay (curve falls towards y = 0). If b = 1 the model is constant — not exponential.
The big idea: To build y = a·bˣ from a context: identify a (the starting amount), identify b (the multiplier per period), then write the model. Substitute to make predictions.
Exponential growth model
A colony of 200 bacteria doubles every hour. Write a model for the number N after t hours, then find N at t = 5.
Step by step
- Write the formula.
- Identify a and b.
- Write the model.
- Substitute t = 5.
Final answer
N = 200 · 2ᵗ. After 5 hours: N = 6400 bacteria.
Exponential decay model
A car worth $24 000 loses 15% of its value each year. Write a model for value V after t years.
Step by step
- Write the formula.
- A 15% loss means 85% remains each year.
- Write the model.
Final answer
V = 24 000 · (0.85)ᵗ
Decay factor = 1 − rate: If something loses 15% per year, the decay factor is b = 1 − 0.15 = 0.85. You keep 85% each year. IB awards a mark for correct identification of b — show this step.
Know your predicted grade
Take timed mock exams and get detailed feedback on every answer. See exactly where you're losing marks.
Try Mock Exams Free7-day free trial • No card required
The big idea: The most common errors: using the percentage rate as b instead of 1 ± rate, and confusing growth factor with decay factor.
Wrong
- b = 0.15 for 15% decay (wrong — use 0.85)
- b = 1.2 for 20% decay (should be 0.80)
- Writing 200 × 2ˣ for decay
- Saying b = 2 means 2% growth
Correct
- 15% decay: b = 1 − 0.15 = 0.85
- 20% decay: b = 1 − 0.20 = 0.80
- Growth: b > 1, decay: 0 < b < 1
- b = 2 means doubling each period (100% increase)
Show the b calculation: Always write b = 1 + rate or b = 1 − rate as a step in your solution. IB awards a mark for identifying b correctly — even if the final answer has an error.
Worked example
Apply the key method from Exponential models in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
The big idea: For y = a·bˣ, the horizontal asymptote is y = 0. As x → +∞ for decay (b < 1), the quantity approaches 0 but never reaches it. If a constant c is added (y = a·bˣ + c), the asymptote shifts to y = c.
| Model | Horizontal asymptote | What it means |
|---|---|---|
| y = 200 · (0.8)ˣ | y = 0 | Value approaches zero but never reaches it |
| y = 200 · (0.8)ˣ + 50 | y = 50 | Value never drops below 50 |
Asymptote in context: If asked 'what does the graph approach as time increases?', state the horizontal asymptote as y = c. Then explain what this means in context (e.g. 'the temperature approaches 20°C but never goes below it').