Skill: Write an equation from a linear graph

A linear equation can be written in the form

$$y=mx+b$$

where:

• $m$ is the slope, or the change in $y$ for each change in $x$.
• $b$ is the $y$-intercept, where the line crosses the $y$-axis.

Example: A graph crosses the $y$-axis at $(0,-1)$ and also passes through $(2,3)$. Write an equation for the line.

1. Find the $y$-intercept.
   The graph crosses the $y$-axis at $-1$, so

$$b=-1.$$

2. Find the slope.
   Use the two points $(0,-1)$ and $(2,3)$:

$$m=\frac{\text{change in }y}{\text{change in }x} =\frac{3-(-1)}{2-0} =\frac{4}{2}=2.$$

3. Substitute $m$ and $b$ into $y=mx+b$.

$$y=2x-1$$

So, the equation of the line is

$$\boxed{y=2x-1}.$$

You can check the equation using the points. When $x=0$, $y=2(0)-1=-1$, and when $x=2$, $y=2(2)-1=3$, matching the graph.