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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.
Concept

Slope

The slope of a line passing through the points (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) is the ratio of the vertical change (Δy)(\Delta y) to the horizontal change (Δx)(\Delta x) between the points. The variable mm is most commonly used to represent slope.

m=ΔyΔxm=\dfrac{\Delta y}{\Delta x}

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 Δy\Delta y and run corresponds to Δx.\Delta x.

This gives the following definition for the slope of a line.

m=ΔyΔx=riserunm=\dfrac{\Delta y}{\Delta x}=\dfrac{\text{rise}}{\text{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.
The sign (positive or negative) of each distance corresponds to the direction of the movement between points. Moving to the right, the run is positive; moving to the left, the run is negative. Similarly, moving up yields a positive rise while moving down gives a negative rise.
Method

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. m=riserun m=\dfrac{\text{rise}}{\text{run}}

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.


1

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.


2

Determine the run

To determine the horizontal distance between points AA and B,B, draw a line from AA to the xx-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 44 units.

3

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 22 units.

4

Write the slope ratio

Now that the rise and run are both known, the slope ratio can be written. Simplify if possible. m=riserun=24=12 m=\dfrac{\text{rise}}{\text{run}} = \dfrac{2}{4} = \dfrac{1}{2}


Concept

Types of Slope

The slope of a line measures how yy changes relative to x.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 00 corresponds with a linear function whose graph is a horizontal line. An undefined slope corresponds with a linear function that is a vertical line.
Rule

Slope Formula

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

m=y2y1x2x1m=\dfrac{y_2-y_1}{x_2-x_1}

Here, (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) are two points on the line.
Exercise

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

Solution
The slope of a line can be calculated using the slope formula. To begin, we'll name the points (x1,y1)(x_1,y_1) and (x2,y2).(x_2,y_2). The names can be applied arbitrarily. We'll say (x1,y1)=(2,1)(x2,y2)=(4,5).\begin{aligned} (x_1,y_1) = (2,1) \\ (x_2,y_2) = (4,5). \end{aligned} To calculate the slope, we'll substitute the given values into the rule and simplify.
m=y2y1x2x1m = \dfrac{y_2-y_1}{x_2-x_1}
m=5142m = \dfrac{{\color{#0000FF}{5}}-{\color{#009600}{1}}}{{\color{#0000FF}{4}}-{\color{#009600}{2}}}
m=42m = \dfrac{4}{2}
m=2m = 2
The line that passes through the points (2,1)(2,1) and (4,5)(4,5) has a slope of 22.
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