Growth vs. Decay

Growth Rate and Decay Rate

If value increases by $r\%$ per period: $f(x) = a(1 + r)^x$

If value decreases by $r\%$ per period: $f(x) = a(1 - r)^x$

Example: A car worth $25,000 depreciates by 15% per year: $V(t) = 25000(1 - 0.15)^t = 25000(0.85)^t$

Identifying Exponential Functions

From a Table

Linear: $+3$ each time. Exponential: $\times 2$ each time.

Compound Interest

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

Special case — continuous compounding: $A = Pe^{rt}$

Half-Life and Doubling Time

Half-life: $A = A_0 \left(\frac{1}{2}\right)^{t/h}$ where $h$ is the half-life

Doubling time: $A = A_0 \cdot 2^{t/d}$ where $d$ is the doubling time

SAT Question Types

Type 1: Interpret the Function

"In $f(t) = 500(1.03)^t$, what does the 500 represent? What does 1.03 represent?"

• 500 = initial value
• 1.03 = 3% growth per period ($b = 1 + r = 1 + 0.03$)

Type 2: Growth/Decay Identification

"Does $y = 200(0.85)^x$ represent growth or decay, and by what percent?"

• Decay (because $0.85 < 1$)
• Rate = $1 - 0.85 = 0.15 = 15\%$ decay per period

Type 3: Evaluate at a Specific Time

"If $P(t) = 1000(1.05)^t$, what is $P(10)$?" $P(10) = 1000(1.05)^{10} \approx 1628.89$

Type 4: Compound Interest

"$5,000 at 4% annual interest compounded quarterly for 3 years" $A = 5000(1 + 0.04/4)^{4 \cdot 3} = 5000(1.01)^{12}$

Common SAT Mistakes

1. Confusing growth rate with growth factor: 5% growth has factor $1.05$, not $0.05$
2. Forgetting the initial value: $f(0) = a$, the initial value is the coefficient
3. Using the wrong formula for compounding: Check what $n$ is (monthly = 12, quarterly = 4, etc.)
4. Confusing linear and exponential — check if the table adds or multiplies
5. Not recognizing half-life pattern: Multiply by $\frac{1}{2}$ each half-life period

Function: $P(t) = 100 \cdot 2^t$

Step 2: Find $P(5)$: $P(5) = 100 \cdot 2^5 = 100 \cdot 32 = 3{,}200$

Answer: $P(t) = 100 \cdot 2^t$; after 5 hours there are 3,200 bacteria.

Answer: 18% annual depreciation; worth approximately $16,541 after 3 years.

A) $b = 7$ → this is 600% growth, not 7% ✗ B) $b = 1.7$ → this is 70% growth, not 7% ✗ C) $b = 0.07$ → this is decay (and extreme decay at that) ✗ D) $b = 1.07$ → this is 7% growth ✓

Answer: D) $f(x) = 100(1.07)^x$

Common mistake: Using $0.07$ or $7$ as the base instead of $1.07$.