Now that we have written equations for linear functions using both slope-intercept form and point-slope form, we can choose which method to use based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function f below.

We are not given the slope of the line, but we can choose any two points on the line to find the slope. Choose [latex](0, 7)[/latex] and [latex](4, 4)[/latex]. We can use these points to calculate the slope.

Now we can substitute the slope and the coordinates of one of the points into point-slope form.

If we want to rewrite the equation in slope-intercept form, we would do the following:

If we wanted to find the equation of the line in slope-intercept form without first using point-slope form, we could have recognized that the line crosses the y-axis when the output value is [latex]7[/latex]. Therefore, b = [latex]7[/latex]. We now have the initial value b and the slope m so we can substitute m and b into slope-intercept form of a line.

The function is [latex]f\left(x\right)=-\dfrac{3}{4}x+7[/latex], and the linear equation would be [latex]y=-\dfrac{3}{4}x+7[/latex].

Example

Write an equation for a linear function f given its graph shown below.

Show Solution

Identify two points on the line such as [latex](0, 2)[/latex] and [latex](–2, –4)[/latex]. Use the points to calculate the slope.

[latex]\begin{array}{l}m & =\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\hfill \\ & =\dfrac{-4 - 2}{-2 - 0}\hfill \\ & =\dfrac{-6}{-2}\hfill \\ & =3\hfill \end{array}[/latex]

 

Substitute the slope and the coordinates of one of the points into point-slope form.

[latex]\begin{array}{l}y-{y}_{1}=m\left(x-{x}_{1}\right)\\ y-\left(-4\right)=3\left(x-\left(-2\right)\right)\\ y+4=3\left(x+2\right)\end{array}[/latex]

 

We can use algebra to rewrite the equation in slope-intercept form.

[latex]\begin{array}{l}y+4=3\left(x+2\right)\hfill \\ y+4=3x+6\hfill \\ y=3x+2 \hfill \end{array}[/latex]

 

This makes sense because we can see from the graph above that the line crosses the y-axis at the point [latex](0,2)[/latex], which is the y-intercept, so [latex]b=2[/latex]

In the following video, we show an example of how to write the equation of a line given its graph.