SAT Math Non-Linear Functions

Quadratics, exponentials, and absolute value — what the SAT tests and how to prepare

Non-linear functions are the harder but predictable part of SAT Math. You'll see three main types: quadratics ( $y = a x^{2} + b x + c$ ), exponentials ( $y = a \cdot b^{x}$ ), and absolute value ( $y = ∣ x ∣$ ). Each has its own toolkit, but the recurring theme is: recognize the type from the graph or context, then find roots, vertices, or interpret in context.

Quadratics get the most attention — quadratic formula, factoring, discriminant. Exponentials show up in growth/decay contexts (compound interest, population). Absolute value is rarest, but knowing that $∣ x ∣ = c$ has two solutions ( $x = c$ or $x = - c$ ) covers most of it.

What the SAT actually tests

Factoring $a x^{2} + b x + c$
Quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
Discriminant: $Δ = b^{2} - 4 a c$ and its sign (one, two, or no real roots)
Vertex of a parabola: $x = - \frac{b}{2 a}$
Exponential $y = a \cdot b^{x}$ — growth ( $b > 1$ ) and decay ( $0 < b < 1$ )
Absolute value equations: $∣ f (x) ∣ = c$ splits into two equations

Key concepts

Factored form

When $y = a (x - p) (x - q)$ , the zeros are $p$ and $q$ . Fastest form for reading roots.

Vertex form

When $y = a (x - h)^{2} + k$ , the vertex is $(h, k)$ . Fastest form for max/min questions.

Discriminant

$Δ = b^{2} - 4 a c$ tells you how many real roots $a x^{2} + b x + c = 0$ has. $Δ > 0$ : two, $Δ = 0$ : one, $Δ < 0$ : none.

Exponential growth

When $y = a \cdot b^{x}$ with $b > 1$ , growth is exponential. Yearly $r$ growth is $y = a (1 + \frac{r}{100})^{t}$ . Decay subtracts.

Worked examples

Example 1

The function $f (x) = x^{2} - 8 x + 12$ has zeros at $x = p$ and $x = q$ . What is $p + q$ ?

Solution

Factor: $x^{2} - 8 x + 12 = (x - 2) (x - 6)$ . Zeros: $p = 2$ , $q = 6$ . Sum: $8$ . (Or by Vieta: sum of roots $= - \frac{b}{a} = 8$ .)

💡 Vieta's formulas are often faster than factoring. Sum of roots $= - \frac{b}{a}$ , product $= \frac{c}{a}$ .

Example 2

A city's population grows exponentially. In 2020 it was 50,000; in 2024 it was 73,205. What is the annual growth rate?

Solution

Model: $P (t) = 50000 \cdot (1 + r)^{t}$ . After 4 years: $73205 = 50000 \cdot (1 + r)^{4}$ , so $(1 + r)^{4} = 1.4641$ . Fourth root: $1 + r \approx 1.1$ , so $r \approx 0.1$ = 10%.

💡 Annual growth rate $r$ goes in $(1 + r)^{t}$ where $t$ is years elapsed.

Common pitfalls

Confusing vertex form with factored form. $(x - 2)^{2} + 3$ has vertex $(2, 3)$ , NOT $(- 2, 3)$ .
Forgetting $\pm$ in the quadratic formula.
Dropping the negative root: $x^{2} = 25$ means $x = 5$ OR $x = - 5$ .
For compound interest — using $r$ as a number (0.1) when the formula wants a percent (10).

Exam strategy

Identify the form first: quadratic? exponential? absolute value? Each has its own shortcut. For quadratics: try factoring before the quadratic formula. Vieta's formulas save seconds when the question asks for sums or products of roots. For exponentials: check whether the base $b$ is greater or less than 1 — that tells you growth or decay instantly.

Frequently asked questions

What are Vieta's formulas?

For $a x^{2} + b x + c = 0$ with roots $x_{1}, x_{2}$ : sum $x_{1} + x_{2} = - b / a$ , product $x_{1} \cdot x_{2} = c / a$ . Let you find sum/product of roots WITHOUT solving.

How do I recognize parabola forms?

Standard: $y = a x^{2} + b x + c$ . Vertex: $y = a (x - h)^{2} + k$ , vertex at $(h, k)$ . Factored: $y = a (x - r_{1}) (x - r_{2})$ , roots at $r_{1}, r_{2}$ .

What is an exponential function?

Form $y = a \cdot b^{x}$ . If $b > 1$ — exponential growth. If $0 < b < 1$ — decay. If $b = 1.05$ — 5% growth per period.

Rational vs quadratic — what's the difference?

Rational: $y = P (x) / Q (x)$ where $P, Q$ are polynomials. Quadratic: degree-2 polynomial. Rationals can have asymptotes — quadratics don't.

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