SAT Math — nonlinear graphs and student-produced responses
Two features of the digital SAT Math section are easy to overlook when practicing with text-only questions. Many Advanced Math questions give a graph rather than an equation, and about a quarter of the Math questions are student-produced responses, where you type your own answer instead of choosing one. This material covers both.
Eight questions come with a drawn graph. You choose the factored rule of a parabola from its intercepts and direction of opening, find a and b in y = a·b^x from two marked points, identify a solution of a line-parabola system where the graphs cross, match an absolute value graph and a square-root graph to their rules, compute an average rate of change on a height-time graph of a kicked ball, count the solutions of f(x) = 1 with a horizontal line across a cubic, and predict a value of an exponential decay from two points.
Four questions concern entering answers. Based on the published directions, you practice writing a mixed number as an improper fraction or a decimal, replacing a fraction that is too long by its decimal equivalent, leaving out commas and dollar signs, and entering just one value when a question has several correct answers. The flashcards summarize these rules, including the character limits for positive and negative answers.
The material offers a quiz of twelve four-option questions, each with an explanation that shows the method and names the slip behind each wrong option, and a set of flashcards that collects the rules to remember. Every graph is drawn to scale, and each question also gives a text description of the graph.
The content is based on the College Board's published skill descriptions for the Advanced Math domain of the digital SAT Math section. It is independent practice and is not produced or endorsed by the College Board.
- Read zeros, vertices, intercepts and marked points from graphs of nonlinear functions
- Match a graph to its equation using key features
- Find solutions of equations and systems from their graphs
- Compute an average rate of change from a graph
- Enter numeric answers in the format the student-produced response directions require
Practice material written by Zestly, based on the College Board's published description of the digital SAT Math section (Advanced Math domain: nonlinear functions and their graphs; student-produced response format), as given on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite. The answer-entry rules are paraphrased from the student-produced response directions in the College Board's digital SAT practice tests (2024).
Sample question
The graph of the quadratic function $y = f(x)$ is shown. Which of the following could define $f$?
- $f(x) = (x - 1)(x + 3)$
- $f(x) = (x + 1)(x - 3)$
- $f(x) = -(x + 1)(x - 3)$
- $f(x) = (x + 1)(x + 3)$
See the answer
$f(x) = (x + 1)(x - 3)$
The graph crosses the $x$-axis at $x = -1$ and $x = 3$, so the factors are $x + 1$ and $x - 3$; it opens upward, so the leading coefficient is positive. Check the $y$-intercept: $(0 + 1)(0 - 3) = -3$, which matches the graph. The rule $(x - 1)(x + 3)$ has zeros $1$ and $-3$ (signs reversed), and $-(x + 1)(x - 3)$ opens downward.
- SAT Math — finding the vertex from any form: −b/2a and completing the square
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- SAT Math — rational expressions: common denominators and polynomial division
- SAT Math — radical and rational equations: extraneous solutions
- SAT Math — polynomial functions: factors, zeros and end behavior
- SAT Math — systems with a nonlinear equation: substitution and counting intersections