The words rise and run are sometimes used to describe the slope of a line, especially when the line is given graphically. Rise corresponds to and run corresponds to
This gives the following definition for the slope of a line.
The slope of a line can be found algebraically using the following rule.
One way to express linear function rules is called slope-intercept form.
Sometimes, is used. In either case, and describe the general characteristics of the line. indicates the slope, and indicates the -intercept. The linear function graphed below can be expressed as because it has a slope of and a -intercept at
Write the equation of the line that passes through the points and
To write the equation of the graph of a line in slope-intercept form,
the -intercept, , and the slope of the line, must be found. The following method can be used. As an example, consider the line shown.
The -intercept is the point where the graph intersects the -axis. From the diagram, it can be seen that the -intercept is
The -coordinate of the -intercept can be substituted into for Here, substituting gives
Next, the slope of the line must be determined. From a graph, the slope of a line can be expressed as where rise gives the vertical distance between two points, and run gives the horizontal distance. To find the slope, use any two points. Here, use the already marked -intercept and the arbitratily chosen
From the lines drawn, it can be seen that the and the Therefore, the slope is
The complete equation of the line can now be written by substituting the value of into the equation from Step 2. Here, substitute