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Finding a Slope-Intercept Equation from a Graph

Equations in slope intercept form are created by using the slope and the $y$-intercept of a line to create an equation that looks like this: $$y = mx + b$$

To build an equation in slope-intercept form, you'll have to use a graph to find both the, well, the slope ($m$) and the $y$-intercept ($b$). Let's walk through the process using this graph:

Finding the Slope

First, we'll have to find the slope ($m$). Follow the steps in this explanation to figure it out, choosing some points on your own.

Right, the slope is $-2$! That's because the rise is $-2$ when the run is $1$.

Finding the $y$-Intercept

You've found the slope of the graph, and you know that our equation-in-progress is $y = -2x + b$.

Next, you have to find the $y$-intercept ($b$). Find the $y$-intercept by looking for where the line crosses the $y$-axis on the graph (in other words, the value of $y$ when $x = 0$.

Right, the $y$-intercept is $3$. Now, add that to our template (we had $y = -2x + b$ after we found the slope).

That's it, you're done! The equation of the line above, written in slope-intercept form, is: $$y = -2x + 3$$

Example Walk Through

There are different ways to find the equation to a line from a graph.

1. Find the slope and $y$-intercept of the line from a graph and plug them into the equation.
2. Use two points to calculate the slope, then plug in and solve for the y-intercept.
3. If you are familiar and comfortable with solving systems of linear equation, simply find two points on the line, form a system of equation and solve.

All lines can be expressed in the form of : $$y=mx +b$$ where $b$ is the $y$-intercept, $m$ is the slope.

To find the equation of the line in the graph, we just need to find the $y$-intercept and the slope.

$y$-Intercept

Simply check where the line intersects with the $y$-axis.

In our example, the $y$-intercept is $4$.

Slope

To find the slope involves a bit more calculation. You can find any two points and calculate the slope: $$\text{slope} =\frac{\text{change in }y}{\text{change in }x}$$

In the example: $$m=\frac{2}{4} = \frac{1}{2}$$

Note: Sometimes people mess up the sign of the slope. If you calculate change of $x$ from point A to point B, then also calculate change of $y$ in the same order. To check if you have made a mistake:

• All decreasing lines have negative slopes
• All increasing lines have positive slopes

The equation for the line is: $$y=\frac{1}{2}x + 4$$

To double check, find coordinate values that correspond to this equation, and graph it! The original graph and your new graph should match.

Sometimes, you may want to find a parallel or perpendicular line to the equation that you've solved for! That's an extension of the concept in this lesson.

(Video) Find the Equation of a Linear Graph

by Corbettmaths

This video by corbettmaths shows you how to find the equation of a line given the graph.

$\large{\text{Summary}}$

We need to find two things to find the equation of a line :

1. The $y$-intercept: where the line crosses the $y$-axis.
2. The slope: the rise over run between two points.

The equation of a line is: $$y=mx+b$$ where $m$ is the slope and $b$ is the $y$-intercept.

The first graph he looks at is:

So first we look where it crosses the $y$-axis. Here it's at the $y$-value of $1$. So the $b$ in our equation becomes $1$. $$y=mx+1$$

Next, we have to find the slope or the rise over run. Starting from the $y$-intercept how many units do we have to rise to get to the next point $(1,3)$? Two units! And how many units do we run to get to this point? One unit! So: $$m=\frac21=2$$ Now we know the slope and intercept, so our equation becomes: $$\color{purple}{y=2x+1}$$

Try it Yourself!

What is the equation of the line below? Hit continue to see the answer.

The line is $\color{#8925AE}{y=-3x-2}$.

To find $3m+b$, we just have to plug in $m$ which is the slope, and $b$ which is the $y$-intercept. We found these values for the equation above. We see $m=-3$ and $b=-2$. $$\begin{align} 3m+b&=3(-3)+(-2)\\ &=-9-2\\ &=\color{#8925AE}{-11} \end{align}$$