Cogito
Integrated Math 1 · Chapter 4 · Lesson 1
Exponential Growth and Decay
Multiplying instead of adding.
12 problems · about 21 minutes · F-LE.A.1, F-LE.A.2
What this lesson teaches
The student builds and interprets exponential models of growth and decay.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsA 30% annual rise. What is the growth factor?
Answer 1.3
Why 1.3.
What distinguishes exponential from linear change?
Answer It multiplies by a fixed factor rather than adding a fixed amount.
Why A constant multiplier.
A 10% rise. Growth factor?
Answer 1.1
Why 1 + 0.10.
A 50% fall. Decay factor?
Answer 0.5
Why 1 − 0.50.
y = 100(2)ˣ at x = 3. What is y?
Answer 800
Why 100 × 8.
Build It Up
The same ideas with more to keep track of.
3 problemsy = 80(0.5)ˣ at x = 3. What is y?
Answer 10
Why 80 halved three times.
y = 200(1.5)ˣ at x = 2. What is y?
Answer 450
Why 200 × 2.25.
Does an exponential eventually overtake any linear growth? 1 yes, 0 no.
Answer 1
Why Given enough time.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each situation by the type of change.
Answer Linear: Adding 50 dollars a week, A tank filling at 3 litres a minute · Exponential: A population growing 3 percent a year, A sample halving every hour
Why Adding the same amount is linear; multiplying by the same factor is exponential.
y = 50(3)ˣ at x = 2. What is y?
Answer 450
Why 50 × 9.
The Factor: A 25% annual rise. What is the growth factor?
Answer 1.25
Why 1.25.
The Decay: A 40% annual fall. What is the decay factor?
Answer 0.6
Why 0.6.