Slope

Lines and Slope 

Definition Let (x1,y1) and (x2,y2) be two points then the slope for those two points is rise/run or                     y2 - y1          m =                                      x2 - x1

Example:  

For the points (1,2) and (5,-3) we have

                    -3 - 2              5
        m  =                     =             
                    5 - 1               4

Definition A line through a point P with slope m is the collection of points Q such that the slope between P and Q is m.

Example:  

Sketch the line through the point (1,-1) with slope  m = 3.  

Solution

We plot the point (1,-1) and rise 3 and run 1 from (1,-1) arriving at 

        (1 + 1,-1 + 3)  =  (2,2)  

Then we connect the dots.

Exercise:  

Sketch the line through the point (2,-3) with slope 1/2.

Special Cases

Vertical lines have an undefined slope and horizontal lines have a zero slope.

Example:  

Without graphing, describe the line through

1. (3,5) and (3,2)
   

2. (1,2) and (-3,2)
   

Solution:  

1. We compute the slope:  
   
                     2 - 5            3
           m =               =  -                 
                     3 - 3            0
   
   which is undefined, hence the line is vertical.
   
   

2. We compute the slope:  
   
                        2 - 2                0
           m =                    =  -         =  0         
                       -3 - 2              -5
       
   
   hence the line is horizontal.
   

Exercise

Determine whether the points (1,1), (2,4), and (7,9) are colinear.

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