Nonlinear Functions | Shu's Tutoring

A function is linear if its rate of change is constant — it gains the same amount with every step in $x$. Anything else is nonlinear. The SAT tests nonlinear functions in three main ways: recognizing them from a graph, recognizing them from a table, and analyzing their behavior. This note covers each.

What makes a function nonlinear

A linear function has a constant slope — $y = mx + b$ — and its graph is a straight line. A nonlinear function has a slope that changes as you move along the graph. A few hints that you're looking at a nonlinear function:

The graph is curved (not a straight line).
The equation contains an exponent on $x$ greater than $1$ (like $x^2$, $x^3$).
The variable is inside a square root, absolute value, or denominator.
The variable is in the exponent (like $2^x$).

Common families of nonlinear functions

Six shapes show up over and over on the SAT. Recognize them at a glance and you can reason about behavior without doing much computation.

Quadratic

$$y = ax^2 + bx + c$$

U-shape (parabola). Has a vertex (min or max).

Cubic

$$y = ax^3$$

S-shape. Goes from one infinity to the other.

Square root

$$y = \sqrt{x}$$

Defined only for $x \geq 0$. Grows quickly then flattens.

Absolute value

$$y = |x|$$

V-shape. Always non-negative output.

Exponential

$$y = a^x$$

Grows ($a>1$) or decays ($0<a<1$). Always positive.

Rational

$$y = \frac{1}{x}$$

Two branches. Undefined at $x = 0$ (vertical asymptote).

Recognizing nonlinear from a table

Without seeing the graph or equation, you can tell a function is nonlinear from a table of values: check the differences. If the $y$-values change by the same amount each time the $x$-value steps up by the same amount, it's linear. If the differences are different, it's nonlinear.

EXAMPLE — IS THIS LINEAR?

$x$	$y$	$\Delta y$
$1$	$2$	—
$2$	$5$	$3$
$3$	$10$	$5$
$4$	$17$	$7$
$5$	$26$	$9$

Differences: $3, 5, 7, 9$. Not constant → nonlinear. In fact, the differences themselves form an arithmetic sequence (gap of $2$ each time) — that's a signature of a quadratic. The function is $y = x^2 + 1$.

Sample SAT-style problems

SAMPLE 1 — IDENTIFY FROM EQUATION

Which of the following defines a nonlinear function?

(A) $f(x) = 3x - 7$
(B) $f(x) = \dfrac{x}{4} + 1$
(C) $f(x) = 2 - 5x$
(D) $f(x) = x^2 - 4$

Only (D) has $x$ raised to a power other than $1$. Answer: (D)

SAMPLE 2 — INTERPRETING EXPONENTIAL CONTEXT

A bacterial culture starts with $200$ cells and doubles every hour. Which function $f(t)$ gives the population after $t$ hours?

(A) $f(t) = 200 + 2t$
(B) $f(t) = 200t^2$
(C) $f(t) = 200 \cdot 2^t$
(D) $f(t) = 200 + 2^t$

"Doubles every hour" means multiplying by $2$ each step. That's exponential growth with base $2$, starting at $200$. Answer: (C) Verify: $f(0) = 200 \cdot 1 = 200$ ✓; $f(1) = 200 \cdot 2 = 400$ ✓; $f(2) = 200 \cdot 4 = 800$ ✓.

SAMPLE 3 — DOMAIN OF A SQUARE-ROOT FUNCTION

What is the domain of $f(x) = \sqrt{2x - 6}$? The expression under the square root must be $\geq 0$: $2x - 6 \geq 0$ $x \geq 3$ Domain: all real $x$ with $x \geq 3$.

SAMPLE 4 — RECOGNIZING FROM A TABLE

The table below shows values of a function $f$. Which type best describes $f$?

$x$	$f(x)$
$0$	$3$
$1$	$6$
$2$	$12$
$3$	$24$

(A) Linear
(B) Quadratic
(C) Exponential
(D) Square root

Each $f(x)$ is exactly $2\times$ the previous one. That's a constant multiplier, not a constant difference. A constant multiplier is the signature of an exponential function: $f(x) = 3 \cdot 2^x$. Answer: (C)

Linear vs Exponential — what to look for in tables

Constant difference ($+5$ each step) → linear
Constant ratio ($\times 2$ each step) → exponential
Differences of differences are constant → quadratic

Common error — confusing growth rate

A function that "increases by more each year" sounds linear because the word "increases" is doing a lot of work. But if it increases by a fixed amount, it's linear; if it increases by a fixed percentage, it's exponential. Watch the language carefully:

"Sales grow by $\$200$ per year" → linear
"Sales grow by $5\%$ per year" → exponential