Cogito
Integrated Math 3 · Chapter 4 · Lesson 1
Exponential Functions and Models
Growth that feeds on itself.
12 problems · about 21 minutes · F-LE.A.1, A-SSE.B.3
What this lesson teaches
The student builds and interprets exponential models including compound growth.
- Exponential change multiplies by a fixed factor each period.
- y = a · bˣ, with b above 1 for growth and below 1 for decay.
- More frequent compounding approaches the continuous limit A = Pe^(rt).
Warm Up
Straightforward practice. Get the method working first.
5 problemsA 15% annual rise. What is the growth factor?
Answer 1.15
Why 1.15.
What distinguishes exponential from linear change?
Answer It multiplies by a fixed factor rather than adding a fixed amount.
Why A constant multiplier.
A 20% rise. Growth factor?
Answer 1.2
Why 1 + 0.2.
A 30% fall. Decay factor?
Answer 0.7
Why 1 − 0.3.
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 problems$200 at 50% growth for 2 years. Final amount in dollars?
Answer 450
Why 200 × 2.25.
160 g halves every hour. Grams after 4 hours?
Answer 10
Why 80, 40, 20, 10.
A population doubles every 5 years. Factor over 15 years?
Answer 8
Why Three doublings.
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: Saving 50 dollars every week, A tank filling at 3 litres a minute · Exponential: A balance growing 6 percent a year, A car losing 15 percent of its value each year
Why A fixed amount against a fixed percentage.
A 25% rise. Growth factor?
Answer 1.25
Why 1 + 0.25.
The Factor: A 6% annual rise. What is the growth factor?
Answer 1.06
Why 1.06.
The Balance: $500 at 10% compounded annually. Balance after 2 years, in dollars?
Answer 605 dollars
Why $605.