Digital SAT Math · Advanced Math

Interpreting nonlinear graphs and models

Most nonlinear items on the Digital SAT never ask you to complete the square or write a growth factor. They hand you a curve — or a short description of one — and ask what a feature means: where the graph starts, how high it gets, when it peaks, over which inputs it is rising, or what happens as time runs long. The algebra lives on the sibling pages; here the work is reading. Label the axes, locate the intercepts and the turning point, translate the question word into a graph feature, and only then compute if the picture will not give the number directly.

On the test

Recognition cues: a curve that turns once (quadratic) or bends without turning (exponential); axis labels with units (seconds, dollars, grams); the words maximum, minimum, intercept, increasing, decreasing, approaches, how many times; and any stem that describes a graph without printing an equation.

What this page owns

The College Board files quadratic and exponential behaviour under one “nonlinear functions” heading. We already split the algebra of those curves onto quadratic functions (three forms, vertex formula, rebuild from intercepts) and exponential functions (percent to multiplier, periods in the exponent). This page owns the picture-first half of the same domain: you are given the graph (or a description of its features) and asked to name, locate, or interpret a feature without rewriting the rule. When the answer is a root of an equation you solve, that is nonlinear equations. When the graph is a straight line, that is linear functions. Form-to-feature algebra — “which equivalent form displays the vertex” — stays on the quadratic page; here the form is the graph itself.

Pattern recognition

Six shapes cover nearly every item:

1. Read a marked point — intercept, vertex, or labeled height is the answer as printed.
2. Value versus location — the question wants either the max/min value (a $y$) or where it occurs (an $x$); both numbers are always nearby.
3. Interval of increase or decrease — the side of the vertex where the curve is rising or falling.
4. Count crossings — how many times the graph reaches a height, meets a horizontal line, or hits zero.
5. End behaviour — what an exponential does as the input grows without bound (approaches a floor; never turns around).
6. Interpret in context — the same feature, restated in the units on the axes (empty tank, peak revenue, launch height).

Method

1. Label the axes in the story’s units. Seconds on the bottom and feet on the side make “when” and “how high” impossible to swap. If the stem is pure $x$ and $y$, still name which axis is the input.
2. Locate the intercepts and the turning point, and say what each means. $y$-intercept = value at input zero. $x$-intercept = input where the output is zero. Vertex = highest or lowest point, and the sign of the opening tells you which.
3. Translate the question word into a feature using the table above. Do this before you compute. Most wrong answers on this skill are a correct number attached to the wrong feature.
4. Read first; compute only if the feature is not labeled. A marked vertex is a free answer. An unmarked height may need one substitution into a printed model, or a symmetry argument from two equal outputs.
5. For intervals, stay on the input axis. Increasing and decreasing describe how outputs change as inputs move — the boundaries are $x$-values (or $t$-values), never the maximum height.
6. For end behaviour, refuse the parabola instinct. An exponential that decays toward the axis does not turn around, does not go negative, and does not “reach” zero at a finite time.

Practice

Answer before opening the explanation. Two items are student-produced response, matching the grid-in share of live nonlinear sets; six hand you a curve because on this skill the graph is the question, not an illustration. Every wrong choice below is one named misread — a value reported where a location was asked, an $x$-intercept treated as a start, the wrong side of a vertex, or a height read off the axis instead of the curve. When you miss one, log the misread, not the item number.

Common mistakes

1. Reporting the maximum value when the question asked where it occurs — the vertex is $(h, k)$; “at what time / price / $x$” wants $h$, and $k$ is always in the choices.
2. Reporting where it occurs when the question asked for the value — the same swap in the other direction; check units (seconds vs feet, dollars of price vs dollars of revenue).
3. Treating an $x$-intercept as the starting value — starting value is the $y$-intercept (output at input zero). An $x$-intercept is when the output is zero.
4. Treating the $y$-intercept as the maximum — only true if the vertex sits on the $y$-axis. Otherwise the intercept is just another point on the way up or down.
5. Judging “increasing” over the wrong interval — for a downward parabola the rising half is left of the vertex; flipping the inequality is the default trap.
6. Writing an interval in terms of the max value — $x > 16$ when the vertex is $(3, 16)$ confuses an output with an input boundary.
7. Counting one intersection whenever a peak exists — a horizontal line below the peak cuts a downward parabola twice; it cuts once only when the line is the peak height.
8. Reading a $y$-value off the axis instead of the curve — the tick mark nearest the point is not the output; follow the vertical line from the input up (or down) to the curve, then across.
9. Expecting an exponential to turn around — end behaviour is a floor or a runaway, never a vertex. “Reaches a maximum and then increases again” is parabola language.
10. Predicting a negative value for a positive decay model — mass, population, and price that decay toward zero stay non-negative.
11. Swapping coordinates in a context sentence — “revenue of \10$at a price of$$500$” is the vertex pair with units reversed.
12. Building a formula when the graph already answers — if the stem shows labeled points, the work is reading and translating, not completing the square.

FAQ

Do I need the equation of the curve? Only when the stem prints one and the feature you want is not labeled. A graph with a marked vertex is already the answer to both “what is the max” and “where does it occur.”

How do I know whether the vertex is a max or a min? Opening direction. Downward (the ends fall away from the turning point) means a maximum; upward means a minimum. Context helps too: a rocket’s height graph that returns to the ground is a downward parabola.

What is the difference between an $x$-intercept and a root? None as points on the graph — both are inputs where the output is zero. The difference is framing: “root” is algebra language, “$x$-intercept” is graph language. In context, that input is when the tank is empty, the rocket lands, or revenue hits zero.

Can a function be increasing on both sides of the vertex? Not if it has a single turning point. A parabola changes monotonicity exactly once. An exponential with base greater than 1 is increasing on its whole domain and has no vertex at all.

How do I count “how many times” without solving? Compare the target height to the peak. Above the peak: zero times. Equal to the peak: once. Strictly between the peak and the lower end of the window: twice for a single-turn parabola that crosses that height on both sides.

Grid-in format? Enter the exact value. The most common grid-in loss on this skill is entering the vertex’s other coordinate — there are no answer choices to warn you that you answered “where” when the item asked “what,” or the reverse.

How is this different from the quadratic-functions page? That page starts from a printed expression and rewrites it. This page starts from a picture (or a description of one) and names a feature. Same curves; different object of the question.

Editorial team · last reviewed 2026-08-12