FIND THE SLOPE OF A LINE FROM THE GRAPH

The formula for slope is referred to rise over run,

Because the fraction consists of the rise (the change in y, going up or down) divided by the run (the change in x, going from left to the right).

The diagram shown below illustrates this. 

The simplest way to look at the slope is

rise / run

(rise over run)

In the formula (rise / run), we can "rise" up or down... but, we ALWAYS "run" to the right.  

Slope of a Line : Positive or Negative or Zero or Undefined

To know the sign of the slope of a straight line, always look at the straight line from left to right. 

(i) When you look at the line, if it goes up, then the line is called rising line and its slope will be a positive value.

(ii) When you look at the line, if the line goes down, then the line is called falling line and its slope will be a negative value.

(iii) If the line is horizontal, the slope will be zero. 

(iv) If the line is vertical, the slope will be undefined. 

Alternate Method :

When the two points (x₁, y₁) and (x₂, y₂) on the line are known, the formula given below can be used to find the slope of the line. 

Solved Examples

Example 1 :

Find the slope of the line shown below. 

Solution :

The above line is a rising line. So, its slope will be a positive value.

Mark two points on the line such that both the x-coordinate and y-coordinate are integers. 

So, we can mark the points (0, 4) and (-3, -4) and measure the rise and run.

For the above line, 

Rise  =  8

Run  =  3

Then, 

Slope  =  rise / run

Slope  =  8/3

Example 2 :

Find the slope of the line shown below. 

Solution :

The above line is a falling line. So, its slope will be a negative value.

Mark two points on the line such that both the x-coordinate and y-coordinate are integers. 

So, we can mark the points (1, -1) and (3, -4) and measure the rise and run.

For the above line, 

Rise  =  3

Run  =  2

Then, 

Slope  =  rise / run

Slope  =  -3/2

Example 3 :

Find the slope of the line shown below. 

Solution :

The above line is an horizontal line. 

So, its slope is zero. 

Example 4 :

Find the slope of the line shown below. 

Solution :

The above line is a vertical line. 

So, its slope is undefined.  

Example 5 :

Find the slope of the line using formula. 

Solution :

Mark two points on the line such that both the x-coordinate and y-coordinate are integers. 

So, we can mark the points (1, -1) and (4, 3) and measure the rise and run.

Formula :

Slope  =  (y₂ - y₁) / (x₂ - x₁)

Substitute (x₁, y₁) = (1, -1) and (x₂, y₂) = (4, 3). 

Slope  =  [3 - (-1)] / (4 - 1)

Slope  =  [3 + 1] / 3

Slope  =  4/3

Example 6 :

Find the slope of the line using formula. 

Solution :

Mark two points on the line such that both the x-coordinate and y-coordinate are integers. 

So, we can mark the points (-1, 4) and (4, -4) and measure the rise and run.

Formula :

Slope  =  (y₂ - y₁) / (x₂ - x₁)

Substitute (x₁, y₁) = (-1, 4) and (x₂, y₂) = (4, -4). 

Slope  =  (-4 - 4) / [4 - (-1)]

Slope  =  -8 / [4 + 1]

Slope  =  -8/5

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