What is the difference between nonlinear equations and nonlinear functions on the SAT?

Equation questions ask you to solve for a value. Function questions ask you to describe behavior: vertex location, intercepts, growth rates, and what a graph or table implies. Same math, different question.

How do I find a parabola's vertex fast?

Three ways: read (h, k) straight from vertex form, compute x = −b/(2a) from standard form, or average the two zeros. In Desmos, just tap the lowest or highest point of the curve.

How do I tell exponential growth from decay?

Look at the base of the exponent. A multiplier above 1 grows, a multiplier between 0 and 1 decays. A base of 0.92 means an 8 percent drop per period.

How many nonlinear function questions will I see?

Roughly 3 to 5 across both modules, since Advanced Math carries about 35 percent of the section and functions are its biggest slice.

Why practice with regenerating questions instead of a fixed list?

A fixed list teaches you its own answers. Regenerating drills swap the vertex, the zeros, and the context every time, so the skill you build transfers to the version the SAT hands you.

Advanced Math · Skill 3 of 3

SAT Nonlinear Functions Practice

Nonlinear functions questions are about reading curves: where a parabola peaks, where a graph crosses zero, how quickly an exponential model grows. The digital SAT mixes equations, graphs, and tables across both of its 35 minute modules. Here is the pattern map, plus drills that refresh forever.

Domain: Advanced Math
About 35% of the test is Advanced Math
Difficulty: easy to hard
Free, no account

Drill nonlinear functions now → The Advanced Math lineup

The patterns every curve question follows

Curves on the SAT behave in predictable ways. Learn these four and the graphs start talking.

Pattern 01

Vertex form and shifts

In y = a(x − h)² + k the vertex sits at (h, k), and the sign of a decides whether the parabola opens up or down. Transformation questions nudge h and k and ask what happened to the graph.

Pattern 02

Zeros and factors

Each factor (x − r) plants an x intercept at r, and the vertex waits exactly halfway between the zeros. A favorite question asks which form of a function displays its zeros as constants.

Pattern 03

Exponential growth and decay

Models take the shape f(t) = a(b)^t, where a is the starting amount and b is the multiplier applied each period. A b of 1.06 grows 6 percent per period; a b of 0.94 shrinks by the same amount.

Pattern 04

Reading the graph

Given a curve, identify where the function increases, the coordinates of its maximum, or how many times it meets a horizontal line. No algebra involved, just careful eyes on the axes.

One model, read start to finish

Worked example · medium

A cafe models its weekly smoothie revenue, in dollars, as R(p) = −20p² + 240p, where p is the price per smoothie in dollars. According to the model, what is the maximum weekly revenue?

Factor to expose the zeros: R(p) = −20p(p − 12), so revenue is zero at p = 0 and p = 12.
The vertex lies midway between the zeros, at p = (0 + 12) ÷ 2 = 6.
The coefficient −20 is negative, so the parabola opens downward and that vertex is a maximum.
Evaluate: R(6) = −20 × 36 + 240 × 6 = −720 + 1,440 = 720. Charging $6 per smoothie yields the peak.

Answer: $720

New cafe, new coefficients, same geometry: zeros first, vertex in the middle, then evaluate.

Misreads that cost easy points

Misreading vertex form signs. The vertex of y = (x − 3)² + 5 is (3, 5), not (−3, 5). The subtraction inside the parentheses hides the sign flip.
Reporting where instead of what. Maximum value questions want the output, 720, not the input that produced it. Both numbers always appear in the choices.
Reversing growth and decay percentages. A multiplier of 0.85 means a 15 percent decrease each period, not an 85 percent one. Subtract from 1 before you interpret.
Shifting the wrong direction. f(x + 4) slides the graph 4 units left, and f(x) + 4 slides it up. Horizontal shifts move opposite to their sign, which is exactly why the SAT asks.

Desmos is your graphing shortcut

Graph the function and its vertex and zeros appear as tappable gray points, no completing the square required. Transformations turn visual too: type f(x) and its shifted cousin together and watch exactly what moved. On this skill Desmos is less a backup and more a first resort.

Why drilling here is different

Curves click after you have seen hundreds of them, and generators make hundreds cheap. Each nonlinear functions drill regenerates its vertex, zeros, and growth rates every time it loads, across three difficulty tiers. All of it flows from a 1,483 question bank in which every answer and explanation has been independently checked.

Learn every curve by heart.

Start graphing free →

Five questions, five answers

What is the difference between nonlinear equations and nonlinear functions on the SAT?: Equation questions ask you to solve for a value. Function questions ask you to describe behavior: vertex location, intercepts, growth rates, and what a graph or table implies. Same math, different question.
How do I find a parabola's vertex fast?: Three ways: read (h, k) straight from vertex form, compute x = −b/(2a) from standard form, or average the two zeros. In Desmos, just tap the lowest or highest point of the curve.
How do I tell exponential growth from decay?: Look at the base of the exponent. A multiplier above 1 grows, a multiplier between 0 and 1 decays. A base of 0.92 means an 8 percent drop per period.
How many nonlinear function questions will I see?: Roughly 3 to 5 across both modules, since Advanced Math carries about 35 percent of the section and functions are its biggest slice.
Why practice with regenerating questions instead of a fixed list?: A fixed list teaches you its own answers. Regenerating drills swap the vertex, the zeros, and the context every time, so the skill you build transfers to the version the SAT hands you.

Round out the domain

Function fluency rests on the two skills that come before it. Close the loop on Advanced Math here.