Write an equation from a graph

A straight-line graph is translated into an equivalent linear equation, typically $y=mx+b$ , by interpreting the slope $m$ as the constant rate of change and the $y$ -intercept $b$ as the value of $y$ when $x=0$ . The reasoning includes reading the graph’s scale, determining slope from two points, and distinguishing the $y$ -intercept from the $x$ -intercept; nonlinear, piecewise, and more abstract forms are outside this scope.

Detailed Explanation: Write an equation from a graph

A straight-line graph can be written as

y=mx+b

where:

$m$ is the slope, or rate of change.
$b$ is the $y$ -intercept, where the line crosses the $y$ -axis.

Example

Suppose a graph shows a line passing through the points $(0,2)$ and $(3,8)$ .

Step 1: Find the $y$ -intercept

The $y$ -intercept is the point where $x=0$ . The graph includes $(0,2)$ , so

b=2

Step 2: Find the slope

Use two points on the line:

m=\frac{\text{change in }y}{\text{change in }x}

From $(0,2)$ to $(3,8)$ :

The change in $y$ is $8-2=6$ .
The change in $x$ is $3-0=3$ .

Therefore,

m=\frac{6}{3}=2

The line rises $2$ units for every $1$ unit it moves to the right.

Step 3: Substitute into $y=mx+b$

Use $m=2$ and $b=2$ :

\boxed{y=2x+2}

Check that $b$ is the $y$ -intercept, not the $x$ -intercept. The $y$ -intercept is where the graph crosses the vertical $y$ -axis.

Learn by doing: Write an equation from a graph

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Slope of a Line - Select Linear Equation Based on Graph

Earned ?