Tables, Graphs, and Equations of Linear Functions — SAT Math

Introduction

Linear functions appear across roughly 35% of SAT Math questions in the Algebra domain, and one of the most-tested skills is moving fluidly between the three representations of the same line: its equation, its table of values, and its graph. On test day you might be asked to identify the equation that matches a graph, find a missing table value, or pick the graph description that fits a table — sometimes in a real-world context, sometimes purely abstract.

This note covers exactly that: reading slope and intercepts from tables and graphs, and translating among all three representations. For interpreting what the slope and intercept mean in a word problem, see Slope and Intercept in Context. For writing a linear function from given conditions or using function notation, see Function Notation and Writing Linear Functions.

Core Concept

Every non-vertical linear function can be written as

y = mx + b

where $m$ is the slope and $b$ is the y-intercept. The same information lives in a table and a graph — you just read it differently.

From a graph

The y-intercept is the point where the line crosses the y-axis; its y-coordinate equals $b$ .
The slope is the ratio $\dfrac{\text{rise}}{\text{run}}$ between any two grid points you can read exactly.

From a table

The y-intercept is the y-value paired with $x = 0$ . If the table doesn't include $x = 0$ , use the slope to work back.
The slope is $m = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2 - y_1}{x_2 - x_1}$ for any two rows. Because the function is linear, you'll get the same slope no matter which two rows you choose.

Quick illustrative calculation

Suppose a table contains the rows $(1,\ 5)$ and $(4,\ 14)$ .

m = \frac{14 - 5}{4 - 1} = \frac{9}{3} = 3

Plug one point into $y = 3x + b$ : $5 = 3(1) + b \Rightarrow b = 2$ .

The equation is $y = 3x + 2$ , and the y-intercept is $(0, 2)$ .

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