Exponential Growth and Decay

What are Exponential Functions?

Exponential function: f(x) = a · b^x

Where:

• a = initial value (when x = 0)
• b = base (growth/decay factor)
• x = exponent (often represents time)

Key difference from linear:

• Linear: Add same amount each step
• Exponential: Multiply by same factor each step

Exponential Growth

Growth occurs when b > 1

Formula: y = a · b^x where b > 1

Characteristics:

• Starts slow, then increases rapidly
• Graph curves upward
• Never touches x-axis (always positive)

Example 1: Population Growth

Initial population: 100 Doubles each year

Formula: P(t) = 100 · 2^t

Table:

After 5 years: P(5) = 100 · 2^5 = 3,200

Example 2: Investment

Invest $1,000 at 5% annual growth

Formula: A(t) = 1000 · (1.05)^t

After 10 years:

A(10) = 1000 · (1.05)^10
A(10) = 1000 · 1.629
A(10) ≈ $1,629

Growth Factor vs. Growth Rate

If growth rate is r (as decimal):

Growth factor: b = 1 + r

Examples:

• 5% growth → r = 0.05 → b = 1.05
• 20% growth → r = 0.20 → b = 1.20
• 150% growth → r = 1.50 → b = 2.50

Example: Find Growth Rate

Formula: y = 500 · (1.08)^t

Growth factor: b = 1.08 Growth rate: r = 1.08 - 1 = 0.08 = 8%

Growing at 8% per time period

Exponential Decay

Decay occurs when 0 < b < 1

Formula: y = a · b^x where 0 < b < 1

Characteristics:

• Decreases rapidly at first
• Approaches zero but never reaches it
• Graph curves downward

Example 1: Radioactive Decay

Initial amount: 80 grams Half-life: Decreases by half each hour

Formula: A(t) = 80 · (0.5)^t

Table:

After 5 hours: A(5) = 80 · (0.5)^5 = 2.5 g

Example 2: Car Depreciation

Car value: $25,000 Depreciates 15% per year

Decay rate: r = 0.15 Decay factor: b = 1 - 0.15 = 0.85

Formula: V(t) = 25000 · (0.85)^t

After 3 years:

V(3) = 25000 · (0.85)^3
V(3) = 25000 · 0.614
V(3) ≈ $15,350

Decay Factor vs. Decay Rate

If decay rate is r (as decimal):

Decay factor: b = 1 - r

Examples:

• 10% decay → r = 0.10 → b = 0.90
• 25% decay → r = 0.25 → b = 0.75
• 5% decay → r = 0.05 → b = 0.95

Compound Interest

Special case of exponential growth

Formula: A = P(1 + r/n)^(nt)

Where:

• P = principal (initial amount)
• r = annual interest rate (decimal)
• n = times compounded per year
• t = years

Example: Compound Interest

Principal: $5,000 Rate: 4% per year Compounded: Quarterly (n = 4) Time: 5 years

Calculate:

A = 5000(1 + 0.04/4)^(4×5)
A = 5000(1.01)^20
A = 5000(1.220)
A ≈ $6,100

Half-Life

Half-life: Time it takes for substance to decay to half its amount

If half-life is h:

After 1 half-life: Amount = (1/2) of original After 2 half-lives: Amount = (1/4) of original After 3 half-lives: Amount = (1/8) of original

Formula: A(t) = A₀ · (1/2)^(t/h)

Example: Medicine in Body

Initial dose: 200 mg Half-life: 6 hours

After 12 hours (2 half-lives):

A(12) = 200 · (1/2)^(12/6)
A(12) = 200 · (1/2)^2
A(12) = 200 · 0.25
A(12) = 50 mg

Comparing Growth and Decay

Exponential vs. Linear Growth

Linear: y = 50x + 100

• Increases by 50 each step

Exponential: y = 100 · 2^x

• Doubles each step

Comparison:

Exponential eventually grows much faster!

Real-World Applications

Biology:

• Bacterial growth
• Population dynamics

Finance:

• Compound interest
• Investment growth

Physics:

• Radioactive decay
• Temperature cooling

Medicine:

• Drug concentration in blood
• Virus spread

Technology:

• Moore's Law (computer power)
• Social media growth

Practice