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How do you recognise and interpret nonlinear functions such as exponential, polynomial, rational and radical functions?

Nonlinear functions: distinguish linear from exponential growth, interpret polynomial, rational, radical and exponential functions and their graphs, and read key features and end behaviour.

A focused answer to the Digital SAT Advanced Math skill of nonlinear functions: telling linear from exponential growth, interpreting exponential, polynomial, rational and radical functions and graphs, and reading their key features in context.

Generated by Claude Opus 4.811 min answerUpdated 2026-06-14

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

Nonlinear functions include exponential ( $f(x) = a \cdot b^x$ ), polynomial (degree $\ge 2$ ), rational (a ratio of polynomials), and radical (involving roots). The headline idea is linear versus exponential growth: a linear function adds a constant amount each step (constant difference), while an exponential function multiplies by a constant factor each step (constant ratio), so exponential growth eventually overtakes any linear growth. In $f(x) = a \cdot b^x$ , $a$ is the initial value and $b$ is the growth factor: $b > 1$ is growth (a factor of $1.05$ is $+5\%$ ), and $0 < b < 1$ is decay (a factor of $0.9$ is $-10\%$ ). Read features from the graph: intercepts, asymptotes, increasing or decreasing intervals, and end behaviour.

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What this skill is asking
Recognising the function type
A worked growth-factor question
Linear versus exponential growth
Reading graphs and key features

What this skill is asking

A nonlinear function is any function that is not a straight line: exponential, quadratic and higher-degree polynomial, rational, or radical. The Digital SAT (Advanced Math domain) tests whether you can recognise the type from an equation, table or graph, interpret its parameters in context, and describe its key features and behaviour. The single most tested distinction is linear versus exponential growth.

Recognising the function type

Match the form to the family.

A worked growth-factor question

Exponential parameters carry precise meaning.

Linear versus exponential growth

This contrast is the most tested idea in the domain. A linear function changes by a constant amount for each unit increase in the input (equal first differences in a table), while an exponential function changes by a constant percentage, that is, it multiplies by a constant factor (equal ratios in a table). The practical consequences: exponential growth starts slowly but eventually dominates any linear function, and exponential decay approaches but never reaches zero (a horizontal asymptote). To tell the two apart from a table, check whether successive outputs have a constant difference (linear) or a constant ratio (exponential).

Reading graphs and key features

The SAT expects you to read key features off a nonlinear graph: intercepts (where it crosses the axes), asymptotes (lines the graph approaches, common for exponential and rational functions), increasing and decreasing intervals, maximums and minimums (for polynomials), and end behaviour (what happens as $x \to \pm\infty$ ). A question might give a graph and ask which equation matches, or give an equation and ask which graph it produces; in both cases, anchor on a couple of distinctive features (an asymptote, an intercept, the direction of growth) rather than checking every point. Desmos is invaluable here: graph the candidate function and compare its features directly.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Digital SAT Math (style)1 marksA population is modeled by

P(t) = 200(1.05)^t

, where

t

is in years. What does the value 1.05 indicate? (A) The population grows by 105% each year (B) The population grows by 5% each year (C) The population starts at 105 (D) The population decreases by 5% each year

Show worked answer →

The correct answer is (B), the population grows by 5% each year.

In an exponential model $P(t) = a \cdot b^t$ , $a$ is the initial amount and $b$ is the growth factor. Here $a = 200$ (starting population) and $b = 1.05$ . A growth factor of $1.05 = 1 + 0.05$ means a $5\%$ increase per year (the $1$ keeps the existing amount, the $0.05$ adds $5\%$ ).

Digital SAT Math (style)1 marksA function

f(x) = 3(2)^x

is compared with a linear function

g(x) = 3 + 2x

. For large positive

x

, which is true? (A) They are always equal (B)

g

grows faster than

f

(C)

f

grows faster than

g

(D) Both are constant

Show worked answer →

The correct answer is (C), $f$ grows faster than $g$ .

$f(x) = 3(2)^x$ is exponential: each step in $x$ multiplies the output by $2$ . $g(x) = 3 + 2x$ is linear: each step adds $2$ . Exponential growth eventually overtakes any linear growth, so for large $x$ , $f$ grows faster. (At $x = 5$ , $f = 96$ but $g = 13$ .)

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Sources & how we know this

Math Specifications — College Board (2024)
What Are Content Domains? — College Board (2024)