Linear vs. Exponential Models — SAT Math

Introduction

When the SAT presents a table of data or a scatterplot, one of the most common tasks is deciding which family of functions — linear, quadratic, or exponential — best models the relationship. This falls under Problem-Solving and Data Analysis, which makes up roughly 15% of SAT Math questions. Getting the model right unlocks the interpretation questions that follow: reading initial values, growth rates, and long-run behavior from an equation or graph.

Scope of this note: constant differences vs. constant ratios in tables; choosing between linear, quadratic, and exponential models; and interpreting graphs of those models. For fitting lines to scatterplots, see Scatterplots and Lines of Best Fit.

Core Concept

The Three Diagnostic Tests

Given a table where x-values increase by a constant amount (e.g., 1, 2, 3, 4, …), check what happens to the y-values:

Test	What to compute	Signal
Linear	Consecutive differences $y_{n+1} - y_n$	All differences are equal
Exponential	Consecutive ratios $y_{n+1} / y_n$	All ratios are equal
Quadratic	Second differences (differences of differences)	All second differences are equal

Why it works:

A linear function $y = mx + b$ increases by $m$ for each unit step → constant first difference.
An exponential function $y = a \cdot b^x$ multiplies by $b$ at each unit step → constant ratio.
A quadratic function $y = ax^2 + bx + c$ has first differences that form an arithmetic sequence, so its second differences equal a constant $2a$ .

Quick Example

$x$	$y$	1st diff	Ratio
0	5	—	—
1	10	5	2.0
2	20	10	2.0
3	40	20	2.0
4	80	40	2.0

First differences are not constant (5, 10, 20, 40), but ratios are all 2 → exponential model: $y = 5 \cdot 2^x$ .

Comparing Linear vs. Exponential Growth

For large enough $x$ , exponential growth always outpaces linear growth, regardless of the starting values. A linear function adds the same amount each step; an exponential function multiplies, so its rate of increase keeps accelerating. This is visible in a graph: the exponential curve eventually rises far above any straight line.

Reading a Graph of a Model

For an exponential model $y = a \cdot b^x$ :

$a$ is the initial value (the y-intercept, at $x = 0$ ).
$b > 1$ : exponential growth; the curve rises and curves upward.
$0 < b < 1$ : exponential decay; the curve falls toward (but never reaches) zero.
The percent change per unit of $x$ is $(b - 1) \times 100\%$ (growth) or $(1 - b) \times 100\%$ (decay).

For a linear model $y = mx + b$ : the graph is a straight line, and the constant slope $m$ is the rate of change per unit of $x$ .

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