Exponential Functions and Equations (SAT Math)

What is an Exponential Function?

General Form: $y = ab^x$

Where:

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

Exponential Growth vs. Decay

Exponential Growth

When $b > 1$ (base greater than 1)

Example: $y = 5(2)^x$

• Starts at 5 (when $x = 0$)
• Doubles each time $x$ increases by 1
• Graph curves upward

Real-world examples:

• Population growth
• Compound interest
• Bacterial growth
• Viral spread

Exponential Decay

When $0 < b < 1$ (base between 0 and 1)

Example: $y = 100(0.5)^x$

• Starts at 100 (when $x = 0$)
• Halves each time $x$ increases by 1
• Graph curves downward

Real-world examples:

• Radioactive decay
• Depreciation
• Cooling (temperature)
• Medicine leaving body

Growth/Decay Factor

Understanding the Base $b$

If quantity increases by $r$% each period:

$b = 1 + r$ (as decimal)

Examples:

• Grows by 10% → $b = 1 + 0.10 = 1.10$
• Grows by 25% → $b = 1 + 0.25 = 1.25$
• Grows by 5% → $b = 1 + 0.05 = 1.05$

If quantity decreases by $r$% each period:

$b = 1 - r$ (as decimal)

Examples:

• Decays by 20% → $b = 1 - 0.20 = 0.80$
• Decays by 15% → $b = 1 - 0.15 = 0.85$
• Decays by 50% → $b = 1 - 0.50 = 0.50$

Common Exponential Formulas

Compound Interest

Formula: $A = P(1 + r)^t$

Where:

• $A$ = final amount
• $P$ = principal (initial amount)
• $r$ = interest rate (as decimal)
• $t$ = time

Example: $1000 at 5% for 3 years A = 1000(1.05)^3 = 1000(1.157625) = \1157.63$

Population Growth

Formula: $P = P_0(1 + r)^t$

Where:

• $P$ = population at time $t$
• $P_0$ = initial population
• $r$ = growth rate
• $t$ = time

Exponential Decay (Half-Life)

Formula: $N = N_0(0.5)^{t/h}$

Where:

• $N$ = amount remaining
• $N_0$ = initial amount
• $t$ = time elapsed
• $h$ = half-life period

Solving Exponential Equations

Method 1: Same Base

If both sides have same base, set exponents equal

Example: $2^{x+1} = 2^5$

Same base (2) → set exponents equal: $x + 1 = 5$ $x = 4$

Method 2: Rewrite with Same Base

Express both sides as powers of same base

Example: $4^x = 8$

Rewrite with base 2: $(2^2)^x = 2^3$ $2^{2x} = 2^3$ $2x = 3$ $x = \frac{3}{2}$

Method 3: Trial and Error (SAT Strategy)

For SAT multiple choice, plug in answer choices!

Example: $3^x = 27$

Test choices:

• $x = 2$: $3^2 = 9$ ✗
• $x = 3$: $3^3 = 27$ ✓

Exponential Properties (Rules of Exponents)

Product Rule

$a^m \cdot a^n = a^{m+n}$

Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7$

Quotient Rule

$\frac{a^m}{a^n} = a^{m-n}$

Example: $\frac{5^7}{5^3} = 5^{7-3} = 5^4$

Power Rule

$(a^m)^n = a^{mn}$

Example: $(3^2)^4 = 3^{2 \cdot 4} = 3^8$

Zero Exponent

$a^0 = 1$ (for $a \neq 0$)

Example: $7^0 = 1$

Negative Exponent

$a^{-n} = \frac{1}{a^n}$

Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Fractional Exponent

$a^{1/n} = \sqrt[n]{a}$

Example: $8^{1/3} = \sqrt[3]{8} = 2$

$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Example: $16^{3/4} = \sqrt[4]{16^3} = (\sqrt[4]{16})^3 = 2^3 = 8$

Key Features of Exponential Graphs

Y-Intercept

When $x = 0$: $y = a \cdot b^0 = a \cdot 1 = a$

Y-intercept is always $a$ (the coefficient)

Horizontal Asymptote

Graph approaches but never touches $y = 0$

Exponential functions never equal zero!

Domain and Range

Domain: All real numbers ($-\infty$ to $\infty$)

Range:

• Growth ($b > 1$): $y > 0$ (positive only)
• Decay ($0 < b < 1$): $y > 0$ (positive only)

Always Increasing or Decreasing

Growth: Always increasing (left to right) Decay: Always decreasing (left to right)

SAT Exponential Strategies

Identify $a$ and $b$

Read the problem carefully:

• Initial value = $a$
• Growth/decay factor = $b$

Convert Percentages

"Increases by 15%" → $b = 1.15$ "Decreases by 30%" → $b = 0.70$

Use Answer Choices

Plug in values to test!

Recognize Common Forms

• $2^x$ → doubling
• $(0.5)^x$ or $(\frac{1}{2})^x$ → halving
• $10^x$ → powers of 10

Check Units

Time units must match rate units!

Common SAT Question Types

Type 1: Find Final Value

"If population grows 8% per year, what's population after 5 years?"

Use: $P = P_0(1.08)^5$

Type 2: Find Growth/Decay Rate

"Population doubles in 10 years, what's annual growth rate?"

Set up: $2P_0 = P_0(1 + r)^{10}$

Type 3: Find Time

"How long until value doubles?"

Set up equation and solve (or test answer choices!)

Type 4: Compare Functions

"Which function grows faster?"

Compare bases: larger base = faster growth

SAT Tips

• Initial value when $x = 0$ is always $a$
• Growth: $b > 1$ (increases by %)
• Decay: $0 < b < 1$ (decreases by %)
• Percent increase of $r$% → multiply by $(1 + r)$
• Percent decrease of $r$% → multiply by $(1 - r)$
• Same base? Set exponents equal
• Can't get same base? Plug in answer choices!
• Graph curves up = growth, curves down = decay
• Horizontal asymptote at $y = 0$ (never touches)
• Doubling = base of 2, halving = base of 0.5
• Compound interest: $A = P(1 + r)^t$