Find the Slope and y-intercept: Mini-Lesson Review

Graph a Line Using Its Slope and $y$-intercepts

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using one point and the slope of the line. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.

Let’s look at the graph of the equation $y=\frac{1}{2}x+3$ and find its slope and $y$-intercept.

The red lines in the graph show us the rise is 1 and the run is 2. Substituting into the slope formula:

$m=\frac{rise}{run}$

$m=\frac{1}{2}$

The $y$-intercept is 3.

Look at the equation of this line.

Look at the slope and $y$-intercepts.

Slope $m$ equals 1 divided by 2 and $y$-intercept (0, 3). The 1 divided by 2 is emphasized in red. The 3 is emphasized in blue.

When a linear equation is solved for $y$, the coefficient of the $x$ term is the slope and the constant term is the $y$-coordinate of the $y$-intercept. We say that the equation $y=\frac{1}{2}x+3$ is in slope–intercept form. Sometimes the slope–intercept form is called the “y-form.”

The slope–intercept form of an equation of a line with slope $m$ and $y$-intercept, (0,b) is $y=mx+b$.

Let’s practice finding the values of the slope and $y$-intercept from the equation of a line.

Example

Identify the slope and $y$-intercept of the line from the equation:

$y=-\frac{4}{7}x-2$