nonlinear functions

Nonlinear functions—quadratic, cubic, rational, radical, absolute value, exponential—appear throughout the SAT. This deep dive gives you graph shape, domain, transformations, intercepts, vertex, and typical SAT question types (roots, maxima, modeling, intersections), with full explanations and hard practice.

what are nonlinear functions?

Nonlinear functions are functions whose graphs are NOT straight lines. On the SAT, these show up as equations, graphs, tables, or word problems. Here are the most common types:

Quadratic:

x^2

Absolute Value:

|x|

Square Root:

\sqrt{x}

Reciprocal:

\frac{1}{x}

Exponential:

2^x

Quadratic (

y = ax^2 + bx + c

U-shaped parabola. Opens up if $a > 0$ , down if $a < 0$ .
Vertex: $x = -\frac{b}{2a}$ . Y-intercept: $c$ .
Domain: all x. Range: $[k, \infty)$ or $(-\infty, k]$ .
SAT tip: Vertex is the "min" or "max." Roots are x-intercepts.

Absolute Value (

y = |x-h| + k

V-shape, sharp corner at vertex ( $h, k$ ).
Domain: all x. Range: $[k, \infty)$ .
SAT tip: Sharp "V" means absolute value!

Cubic (

y = x^3

S-shaped curve. May twist once or twice.
Roots: 1 or 3 real solutions. Domain/Range: all x and y.
SAT tip: Cubics twist, no sharp corners. Look for inflection point.

Square Root (

y = \sqrt{x-h} + k

Starts at ( $h, k$ ), only for $x \geq h$ .
Domain: $x \geq h$ .
SAT tip: No graph left of the starting point.

Reciprocal (

y = \frac{a}{x-h} + k

Two branches, never touches $x = h$ or $y = k$ (asymptotes).
SAT tip: Sudden jumps near $x = h$ or approaches but never hits axis.

Exponential (

y = a b^{x-h} + k

Grows (if $b > 1$ ) or decays (if $0 < b < 1$ ), much faster than lines or parabolas.
Domain: all x. Range: $y > k$ .
SAT tip: “Doubles/halves every…” = exponential. Never touches its asymptote.

\text{Quadratic:}\quad y = ax^2 + bx + c \\ \text{Cubic:}\quad y = x^3 \\ \text{Absolute Value:}\quad y = |x| \\ \text{Square Root:}\quad y = \sqrt{x} \\ \text{Reciprocal:}\quad y = \frac{1}{x} \\ \text{Exponential:}\quad y = 2^x

SAT tip: Recognize the graph’s shape. Nonlinear functions have roots, vertices, or asymptotes that are always clues to the right answer.

modeling with nonlinear functions

Quadratic: Used for area problems, projectile motion (height of an object over time), maximizing/minimizing revenue or profit.
Exponential: Models population growth, compound interest, radioactive decay, and anything that “doubles,” “halves,” or changes by a percentage over regular intervals.
Absolute Value: Describes distance from a point (|x−a|), profit/loss scenarios, and “at least” or “no less than” constraints.

Example (Quadratic):
A ball is thrown upward with height (in feet) given by $h(t) = -16t^2 + 40t + 6$ . What is the maximum height?

t_{\text{vertex}} = -\frac{b}{2a} = -\frac{40}{2(-16)} = 1.25

h(1.25) = -16(1.25)^2 + 40(1.25) + 6 = -25 + 50 + 6 = 31

Maximum height: 31 feet

Example (Exponential):
A population triples every decade. If the initial population is 2,000, what will it be in 30 years?

P = 2000 \times 3^{30/10} = 2000 \times 3^3 = 2000 \times 27 = 54,000

Population after 30 years: 54,000

Example (Absolute Value):
The profit for selling x items is modeled by $P(x) = |2x - 100|$ . What value of x gives zero profit?

|2x - 100| = 0 \implies 2x - 100 = 0 \implies x = 50

Break-even at x = 50 items

Let's practice with a few College Board questions!

Question 1

A rectangle has a length that is 15 times its width. The function $y = (15w)(w)$ represents this situation, where $y$ is the area (in square feet) and $w$ is the width. Which of the following is the best interpretation of $15w$ in this context?

Question 2

The function $h(x) = 2(x - 4)^2 - 32$ is defined above. In the xy-plane, the graph of $y = h(x)$ intersects the x-axis at the points $(0, 0)$ and $(t, 0)$ , where t is a constant. What is the value of $t$ ?

Question 3

The function $f(x) = -8(2)x + 22$ is defined above. What is the y-intercept of the graph of $y = f(x)$ in the xy-plane?

✓