Find slope from two points, a table, a graph, or an equation. | Linear functions | SAT Math

Slope is rise over run, no matter whether you read it from points, a table, a graph, or an equation.

Core Idea

Slope equals $c h a n g e i n 𝑦 c h a n g e i n 𝑥$ . Pick any two points — from a table, a graph, or the equation itself — and compute $𝑚 = 𝑦 2 - 𝑦 1 𝑥 2 - 𝑥 1$ .

Understanding

No matter how the information is presented, slope always comes down to one fraction: rise over run.

From two points or a table: Pick any two rows (or points) and compute $𝑚 = 𝑦 2 - 𝑦 1 𝑥 2 - 𝑥 1$ . Use points with clean numbers to avoid arithmetic errors.

From a graph: Choose two points where the line clearly crosses grid intersections. Count the vertical change (rise) and the horizontal change (run). Up is positive, down is negative; right is positive, left is negative.

From an equation: If it's in slope-intercept form $𝑦 = 𝑚 𝑥 + 𝑏$ , the slope is $𝑚$ . If it's in standard form $𝐴 𝑥 + 𝐵 𝑦 = 𝐶$ , rearrange to slope-intercept, or use $𝑚 = - 𝐴 𝐵$ .

Step by Step

Identify two points with exact coordinates — from the table, graph, or by plugging values into the equation.
Compute $𝑚 = 𝑦 2 - 𝑦 1 𝑥 2 - 𝑥 1$ . Keep track of signs.
Simplify the fraction. A slope of $64$ should be written as $32$ .

Misconceptions

Swapping the order: computing $𝑦 2 - 𝑦 1 𝑥 1 - 𝑥 2$ gives the wrong sign. Keep the same point's coordinates in the same position (first or second) for both numerator and denominator.
Reading rise/run backward on a graph: rise is vertical ( $𝑦$ ), run is horizontal ( $𝑥$ ).
Assuming the slope from standard form $3 𝑥 + 2 𝑦 = 12$ is 3 — it's actually $- 32$ .

Question

Worked Example

A linear function passes through the points $(2, 5)$ and $(6, 17)$ . What is the slope of the function?

Select an answer to see the explanation