Start chapters home Start History history History expand_more Community
Community expand_more
menu_open Close
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
No results
{{ searchError }}
Expand menu menu_open home
{{ courseTrack.displayTitle }}
{{ statistics.percent }}% Sign in to view progress
{{ }} {{ }}
search Use offline Tools apps
Login account_circle menu_open
close expand
Linear Functions

Determining the Slope of a Line

The rate of change of a linear function, which is constant, has a special name — slope. There are different ways to determine the slope of a line given the way in which the function is expressed.



The slope of a line passing through the points (x1,y1) and (x2,y2) is the ratio of the vertical change to the horizontal change between the points. The variable m is most commonly used to represent the slope.

The phrase rise and run is sometimes used to describe the slope of a line, especially when the line is given graphically. Rise corresponds to and run corresponds to

With this information, the definition of the slope of a line can be written in terms of rise and run.

Because the rate of change of a linear function is constant, any two points on the line can be used to find the slope – either algebraically with the Slope Formula or graphically as shown below.
The sign of each distance corresponds to the direction of the movement between the points. Moving to the right, the run is positive. Conversely, moving to the left, the run is negative. Similarly, moving up yields a positive rise while moving down gives a negative rise.


Determine the Slope of a Line from its Graph

The slope of a line measures the change between points on the line. When presented graphically, it's common to use the following definition of slope.

Here, rise represents the vertical distance between points and run represents the horizontal distance. Consider the line shown. The slope of the line will be found using the following method.


Mark two points on the line

To begin, mark any two points on the line. It can be helpful to mark points with integer coordinates to be as precise as possible.

Next, it is necessary to determine the rise and run between the points. This can be done in either order.


Determine the run

To determine the horizontal distance between points A and B, draw a line from A to the x-coordiante of B. Then, count the number of steps the segment spans. Remember, moving to the right yields a positive run, while moving to the left yields a negative run.

It can be seen that the run spans 4 units.


Determine the rise

The rise between points can be found in the same way. Remember, moving up yields a positive rise, while moving down yields a negative rise.

The rise spans 2 units.


Write the slope ratio
Now that the rise and run are both known, the slope ratio can be written. Simplify if possible.


Types of Slope

The slope of a line measures how y changes relative to x. There are different types of slope depending on the direction of the line.

Positive, negative, undefined, and no slope
If the slope is positive, it means that the function is increasing, while a negative slope means it is decreasing. A slope of 0 corresponds with a linear function whose graph is a horizontal line. An undefined slope corresponds with a linear function that is a vertical line.


Slope Formula

The slope of a line can be found algebraically using the following rule.

Here, (x1,y1) and (x2,y2) are two points on the line.

What is the slope of the line that passes through the points (2,1) and (4,5)?

Show Solution
The slope of a line can be calculated using the slope formula. To begin, we'll name the points (x1,y1) and (x2,y2). The names can be applied arbitrarily. We'll say
To calculate the slope, we'll substitute the given values into the rule and simplify.
The line that passes through the points (2,1) and (4,5) has a slope of 2.
{{ 'mldesktop-placeholder-grade-tab' | message }}
{{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!
{{ grade.displayTitle }}
{{ 'ml-tooltip-premium-exercise' | message }}
{{ 'ml-tooltip-programming-exercise' | message }} {{ 'course' | message }} {{ exercise.course }}
{{ focusmode.exercise.exerciseName }}
{{ 'ml-btn-previous-exercise' | message }} arrow_back {{ 'ml-btn-next-exercise' | message }} arrow_forward
arrow_left arrow_right