Math · Advanced Math

Nonlinear Functions on the Digital SAT

Nonlinear Functions questions test whether you can read, evaluate, and interpret quadratic and exponential functions. That includes function notation like f(3), finding a parabola's vertex and intercepts, and telling percent-based exponential growth apart from steady linear change. It is the single most common Advanced Math skill on the Digital SAT, so it shows up in both modules.

This is also a skill where Advanced Math separates high scorers from everyone else, and not because the algebra is brutal. The questions reward understanding what each piece of a function means: what the 500 and the 1.02 each do in f(t) = 500(1.02)^t, or why vertex form hands you the minimum value for free. Students who only memorize procedures get picked off by interpretation questions.

If you keep missing these, the fix is usually vocabulary, not computation. Learn what vertex, intercept, initial value, and growth factor each look like inside an equation, and most of these questions turn into reading exercises with a little arithmetic at the end.

New to the Digital SAT? Start with how the test is structured and scored.

What the SAT actually tests

Evaluating functions with function notation, including composed or shifted inputs like f(x + 2) and g(f(1))
Finding a parabola's vertex, axis of symmetry, and minimum or maximum value from standard form or vertex form y = a(x - h)² + k
Finding x-intercepts and y-intercepts of quadratic functions and connecting them to factored form
Reading higher-degree polynomial graphs: counting distinct x-intercepts from factored form (a repeated factor is one intercept) and reading end behavior from the leading term's sign and degree
Exponential growth and decay models of the form f(t) = a(b)^t, where a is the initial value and b is the growth or decay factor
Reading exponential graphs: horizontal asymptote, y-intercept, and end behavior of a shifted model f(x) = a(b)^x + d, where the asymptote is y = d and the y-intercept is a + d
Distinguishing linear change (adds the same amount each step) from exponential change (multiplies by the same factor each step)
Interpreting what constants and coefficients mean in a real-world model, like a starting population or a percent increase per year

Strategies that work

Translate the function into a sentence

For a model like f(t) = 500(1.02)^t, say what each number does: start at 500, multiply by 1.02 each year. The multiplier 1.02 means a 2% increase per year, while 0.98 would mean a 2% decrease. Interpretation questions are testing exactly this sentence, so build it before reading the choices.

Use the vertex, not a table

For any minimum or maximum question on a quadratic, go straight to the vertex: x = -b/(2a) in standard form, or read (h, k) directly from vertex form. Then evaluate the function at that x to get the actual min or max value. Plugging in a list of x values and hoping is slower and misses non-integer vertices.

Substitute the exact input, parentheses and all

To evaluate f(4) for f(x) = x² + 3x, replace every x with (4) before simplifying: (4)² + 3(4). The parentheses keep exponents applied to the whole input and protect against sign errors with negative inputs. Most function notation misses are substitution slips, not concept gaps.

Test the pattern: add or multiply?

Given a table or description, check consecutive outputs. If they go up by the same amount each step, the model is linear; if each output is the previous one times the same factor, it is exponential. This one check answers every "which model fits" question in the skill.

Mistakes to avoid

Treating exponential growth as linear: a population growing 2% per year multiplies by 1.02 each year, it does not add a flat 2 people. Confusing the two is the most common miss on model questions.
Answering with the vertex's x-coordinate when the question asks for the minimum or maximum value, which is the y-coordinate at the vertex.
Sign errors completing the square or reading vertex form: in y = 2(x - 3)² + 5 the vertex is (3, 5), not (-3, 5), and factored-out coefficients multiply the added constant.
Evaluating x² as 2x when substituting, so f(4) for x² + 3x becomes 8 + 12 = 20 instead of 16 + 12 = 28.
Mixing up a and b in f(t) = a(b)^t, like reading the initial value from the growth factor or reporting b = 1.02 as a 102% increase.

Solve it with Desmos

Let the calculator do the work

The Digital SAT's built-in Desmos calculator can crack Nonlinear Functions questions in a few clicks. Here are the moves that apply, step by step.

All 15 Desmos techniques

Practice with Nova, your Math coach

Answer right here and Nova walks you through it, just like in a drill. Difficulty labels are relative within this skill.

Tip: the Calculator button on each question opens the same Desmos graphing calculator you get on the real Digital SAT. Try solving these with it.

Question 1Easy

The function f is defined by f(x) = x² + 3x.

What is the value of f(4)?

Pick an answer to get instant coaching from Nova.

Question 2Medium

The function f(t) = 500(1.02)^t models the population of a town t years after 2020.

Which of the following best describes how the population changes over time?

Pick an answer to get instant coaching from Nova.

Question 3Hard

The function f is defined by f(x) = 2x² - 12x + 23. The graph of y = f(x) is shown in the xy-plane.

y = f(x)

Which of the following is an equivalent form of f(x) that displays the minimum value of f as a constant?

Pick an answer to get instant coaching from Nova.

Keep missing Nonlinear Functions questions?

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The rest of Advanced Math

Nonlinear Functions is one of 3 skills in Advanced Math. The ideas build on each other, so it pays to learn them as a set rather than in isolation. See how Advanced Math fits together.

Other Math skills to explore

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