Slope

The slope of a line passing through the points (x_1,y_1) and (x_2,y_2) is the ratio of the vertical change — the change in y-values — to the horizontal change — the change in x-values — between the points. The variable m is most commonly used to represent the slope, and the variable k is also used.

The phrase rise over run is commonly used to describe the slope of a line, especially when the line is given graphically. Rise refers to change in y-value and run refers to change in x-value. The right triangle used to visualize the rise and run between two points on the graph of a line is called slope triangle.

The definition of the slope of a line can be written in terms of its rise over its run.

The sign of each distance matches the direction of the movement between the points. The run is positive if the direction moves right. Whereas, the run is negative if the direction moves left. Similarly, moving upward results in a positive rise, while moving downward results in a negative rise.

Extra

Interactively Explore Directions of Rise and Run

The following applet shows how the directions of the rise and run affects the function graph. Change the direction by pushing ⬆️, ⬇️, ➡️, and ⬅️ buttons.

In most cases, the run’s direction is considered as moving to the right. Recall that the direction of the rise determines the sign of the slope. In addition, the sign of the slope is the product between the signs of the rise and run.