These Digital SAT Advanced Math questions involve quadratic, exponential, and other nonlinear relationships — identify the family, read key features from the right form, interpret parameters in context, and connect equations, tables and graphs.
Nonlinear functions is one of three skills College Board groups under Advanced Math, next to equivalent expressions and nonlinear equations and systems. Advanced Math is about 35% of SAT Math, or 13 to 15 questions, and those three skills share that count. The equations guide covers solving. This skill is about the function itself: what its equation, graph and table tell you, which form shows a feature, and what each number means in a real situation.
The framework’s description of the skill opens with “Create and use quadratic or exponential functions to solve problems in a variety of contexts.” It also covers interpreting “an input/output pair including an intercept or initial value,” making “connections between a table, an algebraic representation, or a graph,” and choosing “the most suitable form of the expression representing the output of the function to display key features.” Polynomial, simple rational and radical functions appear in the list too.
Math is two adaptive 35-minute modules of 22 questions. About 25% of the questions are student-produced response (SPR), where you type the answer, and about 30% are set in context. College Board’s sample SPR question for this skill describes a function that “decreases by 80%” for each increase in x by 1. That means a factor of 0.2, not 0.8, and the key accepts the answer as 3.44 or 86/25.
College Board’s Skills Insight ties skill statements to Math score bands. These are the ones that describe nonlinear functions, quoted without the notes that limit some statements to particular tests.
Low bands ask you to read a feature or plug in a number, middle bands ask what a number means, and top bands ask you to move between forms, graphs and tables. Sections 04 to 06 cover those moves.
Wrong family, wrong plan. A constant add each step is linear; a constant multiply is exponential; an x² term is quadratic. Name the family first, then use that family’s tools.
Use Name → Interpret → Evaluate/Solve. For f(x) = a(x − h)² + k, vertex is (h, k). For f(x) = ab^x, a is the start and b is the growth/decay factor (b > 1 growth, 0 < b < 1 decay). When a question makes you solve, the methods are in our nonlinear equations guide; for exponentials, you’ll usually evaluate or compare factors rather than “solve for t.”
If f(x) = 3(2)^x, what is f(3)?
For vertex form, read (h, k) cold. For standard ax² + bx + c, vertex x-coordinate is −b/(2a). Factor when integers are friendly. On MC exponential questions, compute one more year by multiplying the previous value by b — don’t restart from a every time if the table already gives a nearby year.
What is the vertex of y = (x − 3)² + 5?
A quadratic function can be written three ways, and each one puts a different feature on display. That’s the framework’s “most suitable form” idea, and it’s the top-band statement in the ladder above.
f(x) = ax² + bx + c shows the y-intercept, because f(0) = c. The sign of a tells you the shape: positive opens up (a minimum), negative opens down (a maximum).f(x) = a(x − r)(x − s) shows the x-intercepts, (r, 0) and (s, 0). The axis of symmetry is halfway between them.f(x) = a(x − h)² + k shows the vertex (h, k). The minimum or maximum value of f is k, and it happens at x = h.Step through one function in all three forms. Notice the shortcut in steps 3 and 4: once you know the x-intercepts, average them to get the vertex’s x-coordinate and plug in. You don’t need to complete the square.
Watch the wording. “The minimum value of f” is an output, k. “The value of x at which f reaches its minimum” is h. College Board’s sample question for this skill includes the x-value as a wrong choice.
The function f is defined by f(x) = 2(x − 1)(x − 7). Which equivalent form of f displays the minimum value of f as a constant or coefficient?
Write any exponential model as f(t) = a · b^(t/k). a is the value at t = 0. b is what you multiply by each period. k is the length of the period (often 1, so it disappears). Get those three from the words and the model writes itself.
1 + r/100; a decrease of r% is 1 − r/100. So 6% growth is 1.06 and a 30% drop is 0.7. Our percentages guide drills this conversion.2^(t/3). Test it: at t = 3 the exponent should be exactly 1.The value of a car, in dollars, t years after it was bought is V(t) = 24,000(0.85)^t. Which statement best describes the meaning of 0.85?
A town’s population t years from now is modeled by P(t) = 5,000(1.2)^(t/4). By what percent does the population increase every 4 years?
Three facts connect a function’s equation to its graph. A point (a, b) on the graph of y = f(x) means f(a) = b. The x-intercepts are the inputs where f(x) = 0. The y-intercept is (0, f(0)); on an exponential, plug in 0 first, since b⁰ = 1.
Translations move the whole graph without changing its shape:
f(x) + k moves the graph up k units (down if k is negative).f(x − h) moves it right h units, and f(x + h) moves it left. The sign inside works the opposite way from how it looks.−f(x) flips it over the x-axis, so a minimum becomes a maximum.Every point moves the same way, vertex included. Example: where is the vertex of g(x) = f(x + 1) − 5 if f(x) = x² − 4x + 7? First put f in vertex form: x² − 4x + 4 + 3 = (x − 2)² + 3, so f’s vertex is (2, 3). The + 1 inside moves it 1 left and the − 5 moves it 5 down, so g’s vertex is (1, −2). Check by substituting: g(x) = (x + 1 − 2)² + 3 − 5 = (x − 1)² − 2 ✓.
The function f is defined by f(x) = (x − 4)² + 1. The graph of y = g(x) is the graph of y = f(x) shifted 3 units to the left and 2 units down. What is the vertex of the graph of g?
The built-in Desmos calculator is available for the whole Math section, and function questions are where it saves the most time.
f(x) = 2(x − 1)(x − 7) and click the parabola. Desmos marks the intercepts and the vertex as gray points you can click for coordinates. Type f(3) on a new line to evaluate.g(x) = f(x + 3) − 2 under f and see where the graph lands, instead of trusting your memory of which way the sign goes.y₁ ~ a·b^(x₁) (exponential) or y₁ ~ ax₁² + bx₁ + c (quadratic). Desmos reports the constants; check them against one row.f(x) = a(x − 3)² + 2 must pass through (5, 10), drag a slider for a until the curve hits the point, then confirm: 10 = 4a + 2, so a = 2.Desmos shows decimals, so it’s slow when the choices are exact forms like 2(x − 4)² − 18, and it can’t tell you what 0.85 means in a story. Use it for numbers; use the method for forms and meaning.
Growth of 5% is factor 1.05, not 0.05 and not 5. Decay of 20% is factor 0.8. For quadratics, forgetting that (x − 3)² opens the same way as x² but shifted. Mixing linear “add 5 each year” with exponential “×1.05 each year” is the story trap. Wrong choices on this skill usually come from one of these slips:
2^(t/3), not 2^(3t) or 3 · 2^t.f(x + 2) moves the graph left, not right.f(0) is where the graph meets the y-axis. The zeros of f are where it meets the x-axis.Classify each model. The last three are tables with x = 0, 1, 2, 3: check differences, then ratios.
Classify the model.
A population decays by 10% per year from 500. Which model matches after t years?
Read whether the quantity is multiplied or added each period. Projectile and area problems are usually quadratic; compound interest and population are exponential; constant-speed distance is linear.
In a height model like h(t) = −16t² + 48t + 6, each feature of the graph has a meaning. The constant is h(0), the height at launch. The vertex gives the greatest height and the time it’s reached. The positive x-intercept is when the object hits the ground. The 610–670 statement in the ladder is about exactly these connections, so name the feature before you calculate anything.
A ball is thrown upward. Its height, in feet, t seconds after it is thrown is h(t) = −16t² + 48t + 6. What is the best interpretation of the number 6?
A ball’s height is h(t) = −5t² + 20t. At what t does it return to the ground (h = 0), besides t = 0?
Answer each one on paper before you reveal it. Once the forms are second nature, most should take only a few seconds.
f(x) = 2^x. Find f(5).
Vertex of y = (x + 2)² − 7
x-intercepts of f(x) = (x − 2)(x − 3)
5% growth factor
Growth factor for a 12% decrease
f(x) = 3(x + 1)² − 8. Vertex? Minimum or maximum?
f(x) = −2(x − 5)(x + 1). Axis of symmetry?
P(t) = 300(2)^(t/6). How long does P take to double?
y-intercept of f(x) = 4(3)^x − 2
(2, 5) is on the graph of y = f(x). Name a point on y = f(x − 3) + 1.
Here are three Nonlinear Functions questions from our problem sets, from easier to harder, in the same format as the Digital SAT. Answer each one to see the full explanation.
Shorter questions on the same skills, from easier to harder. Questions 6 and 7 make you build the function from a table and from two points.
If g(x) = 5(1/2)^x, what is g(2)?
What is the minimum value of y = (x − 1)² + 8?
The function f is defined by f(x) = x² − 49. At which points does the graph of y = f(x) cross the x-axis?
Investment of $1,000 grows by 8% per year. Value after 1 year?
Which function is exponential?
For an exponential function f, f(0) = 6, f(1) = 9 and f(2) = 13.5. Which equation defines f?
The function g is defined by g(x) = a(b)^x, where a and b are positive constants. If g(0) = 5 and g(2) = 45, what is g(3)?
Related: Nonlinear equations · Equivalent expressions · Linear functions · Percentages · SAT Math formulas · SAT Math overview