Slope-intercept form: write an equation from a graph

Key Notes:

Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is:

y = mx + b

where:

  • m = slope (rate of change)
  • b = y-intercept (where the line crosses the y-axis)

Identifying the Slope (m) from a Graph

Slope formula:

m=rise / runm

  • Rise = vertical change (difference in y-values)
  • Run = horizontal change (difference in x-values)

Steps to find slope:

  1. Pick two points on the line (make sure they are exact grid points).
  2. Count how many units you move up/down (rise) and left/right (run).
  3. Divide rise by run.

  • Positive slope → line goes up from left to right.
  • Negative slope → line goes down from left to right.

Finding the y-Intercept (b)

  • Locate the point where the line crosses the y – axis.
  • The y-coordinate of this point is b.
  • Example: If the line crosses at (0, 4), then b = 4.

Writing the Equation

  1. Write the general slope-intercept form: y = mx + b.
  2. Substitute m and b with the values from the graph.
  3. Simplify if needed.

Example

From the graph:

  • Line passes through points (0, 2) and (3, 5).
  • Slope:

m=5−2 / 3−0 = 3 / 3=1

  • y-intercept: b = 2

Equation:

y = 1x + 2 or y = x + 2

Tips

  • Always choose points where the grid lines meet exactly.
  • Check the slope sign carefully (positive or negative).
  • If the line is horizontal: m=0 → equation is y=b.
  • If the line is vertical: slope is undefined → equation is x=a (not in slope-intercept form).

Learn with an example

Look at this graph

What is the equation of the line in slope-intercept form?

solution:

First, find the slope m. Pick two points on the line, such as (0, 3) and (7, 4). Plug those points into the slope formula.

m= change in y/change in x

m = 4-3/7-0

m = 1/7

The slope is =1/7

Now find the y-intercept. Since the line crosses the y-axis at 3, the y-intercept is 3.

Finally, use the slope m =1/7 and the y-intercept b = 3 to write the equation in slope-intercept form.

y = mx + b

y =1/7x + 3

Look at this graph.

What is the equation of the line in slope-intercept form?

First find the slope m. Pick two points on the line, such as (0, -2) and (1, 0). Plug those points into the slope formula.

m = change in y / change in x

m = 0 – -2 / 1 – 0 Plugin (0, -2) and (1, 0)

m = 2/1 Simplify

m = 2 Simplify

The slope is 2.

Now find the y-intercept. Since the line crosses the y-axis at -2, the y-intercept is -2.

Finally, use the slope m = 2 and the y-intercept b = 2 to write the equation in slope-intercept form.

y = mx + b

y = 2x + –2 Plugin m=2 and b = –2

y = 2x − 2 Rewrite as subtraction

Look at this graph.

What is the equation of the line in slope-intercept form?

First find the slope m. Pick two points on the line, such as (0, 10) and (1, 5). Plug those points into the slope formula.

m =change in y/ change in x

m =5 − 10 /1 − 0 Plugin (0, 10) and (1, 5)

m =–5/1 Simplify

m = –5 Simplify

Theslopeis–5.

Now find the y-intercept. Since the line crosses the y-axis at 10, the y-intercept is 10.

Finally,usetheslopem = –5andthey-interceptb = 10towritetheequationinslope-interceptform.

y = mx + b

y = –5x + 10 Plugin m = –5 and b = 10

Let’s Practice!