Exponentials and Nonlinear Systems Strategy

Exponential Growth and Decay

The SAT uses exponential functions to model populations, investments, and radioactive decay. The general form is:

$f(t) = a \cdot b^t$

• $a$ = initial value (when $t = 0$)
• $b$ = growth/decay factor
  • $b > 1$: growth (e.g., $b = 1.05$ means 5% growth per period)
  • $0 < b < 1$: decay (e.g., $b = 0.85$ means 15% decay per period)

The Interpretation Question

The SAT loves asking: "A population is modeled by $P = 1{,}200 \cdot (1.03)^t$, where $t$ is in years. What does $1.03$ represent?"

Answer: Each year, the population is multiplied by $1.03$ — a 3% annual increase.

Common trap: Distractors will say "the population increases by 1.03 each year" (additive, not multiplicative) or "the population is 103" (misreading the base).

Nonlinear Systems

A nonlinear system pairs a linear equation with a quadratic (or two quadratics). The SAT typically asks for the number of intersection points or the coordinates.

Strategy: Substitute the linear equation into the quadratic, then solve the resulting quadratic.

Example: $y = x + 1$ $y = x^2 - 3x + 5$

Substitute: $x + 1 = x^2 - 3x + 5$

Rearrange: $x^2 - 4x + 4 = 0$

Factor: $(x - 2)^2 = 0$, so $x = 2$, $y = 3$.

The system has exactly one solution — the line is tangent to the parabola.

Rational Expressions

The SAT tests your ability to simplify, add, and solve rational expressions. Key skills:

1. Factor before simplifying: $\frac{x^2 - 9}{x + 3} = \frac{(x-3)(x+3)}{x+3} = x - 3$ (when $x \neq -3$)
2. Extraneous solutions: When you multiply both sides by a variable expression, always check that your solution doesn't make a denominator zero.
3. Adding fractions: Find LCD, combine, simplify.

SAT Pattern: "What value of $x$ satisfies $\frac{2}{x-1} + \frac{3}{x+1} = \frac{10}{x^2 - 1}$?"

Since $x^2 - 1 = (x-1)(x+1)$, multiply through by $(x-1)(x+1)$:

$2(x+1) + 3(x-1) = 10$

$2x + 2 + 3x - 3 = 10$

$5x - 1 = 10$

$x = 2.2$

Check: denominators are $1.2$, $3.2$, and $3.84$ — none are zero, so valid.