Exponential Models
Exponential functions model growth and decay processes including population growth, compound interest, radioactive decay, and cooling, using the form y = ab^x or y = ae^(kx).
What You Need to Know
Key Concept Diagram
Exponential growth: y = ab^x where b > 1; the quantity increases at an increasing rate
Exponential decay: y = ab^x where 0 < b < 1; the quantity decreases toward zero
Compound interest uses A = P(1 + r/n)^(nt)
The natural exponential function y = e^x has base e ≈ 2.718
Half-life is the time for a quantity to halve; used in radioactive decay
Key Vocabulary
Exponential function
A function of the form y = ab^x where b > 0 and b not equal to 1
Half-life
The time taken for a quantity to reduce to half its original amount
Compound interest
Interest calculated on both the principal and accumulated interest
Asymptote
A line that the curve approaches but never reaches; exponential functions have a horizontal asymptote
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
A population of 1000 grows at 5% per year. After 2 years the population is:
Question 2
An exponential decay function has which property?
Question 3
$2000 is invested at 4% p.a. compounded annually. After 3 years the amount is:
Key Concepts Summary
- ●Exponential growth: y = ab^x where b > 1; the quantity increases at an increasing rate
- ●Exponential decay: y = ab^x where 0 < b < 1; the quantity decreases toward zero
- ●Compound interest uses A = P(1 + r/n)^(nt)
- ●The natural exponential function y = e^x has base e ≈ 2.718
- ●Half-life is the time for a quantity to halve; used in radioactive decay