Skill: 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.