Exponential Functions and Equations on the SAT

What Is an Exponential Function?

An exponential function has the form: $f(x) = a \cdot b^x$

Where:

• $a$ = initial value (when $x = 0$)
• $b$ = base (growth/decay factor)
• $x$ = typically time or number of periods

---

Growth vs. Decay

| Condition | Type | Example | |---|---|---| | $b > 1$ | Exponential Growth | Population doubling | | $0 < b < 1$ | Exponential Decay | Radioactive decay | | $b = 1$ | No change (constant) | — |

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

| Feature | Linear | Exponential | |---|---|---| | Pattern | Add a constant | Multiply by a constant | | Equation | $y = mx + b$ | $y = a \cdot b^x$ | | Rate of change | Constant | Changes (accelerating) | | Graph | Straight line | Curved |

From a Table

| $x$ | $y$ (Linear) | $y$ (Exponential) | |---|---|---| | 0 | 5 | 5 | | 1 | 8 | 10 | | 2 | 11 | 20 | | 3 | 14 | 40 |

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

---

Compound Interest

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

| Variable | Meaning | |---|---| | $A$ | Final amount | | $P$ | Principal (initial) | | $r$ | Annual interest rate (decimal) | | $n$ | Times compounded per year | | $t$ | Time in years |

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