﻿ Slope

Slope

Lines and Slope

 DefinitionLet (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. 