SLOPE OF LINES
essential concepts

‘slope’ is a number that measures the ‘steepness’ of a line;
it is conventionally denoted by the variable $\,m\,$

Let $\,(x_1,y_1)\,$ and $\,(x_2,y_2)\,$ be two points on a nonvertical line.
Then:
$$
\cssId{s12}{m = \text{slope}}
\cssId{s13}{= \frac{\text{rise}}{\text{run}}}
\cssId{s14}{= \frac{y_2y_1}{x_2x_1}}$$

‘rise’ denotes vertical travel:
 up is positive
 down is negative

‘run’ denotes horizontal travel:
 to the right is positive
 to the left is negative

to find the slope of any nonvertical line:
 choose any two points

travel from one to the other, going vertically first (the rise)
and horizontally second (the run)
 take the rise, and divide by the run
 horizontal lines have slope zero ($m = 0$)
 vertical lines have no slope (the slope does not exist, due to division by zero)
 two nonvertical lines are parallel if and only if they have the same slope

perpendicular lines have slopes that are opposite reciprocals;
for example, lines with slopes $\,3\,$ and $\,\frac 13\,$ are perpendicular

going from left to right:
 uphill lines have positive slopes
 downhill lines have negative slopes
 steep lines have big slopes (numbers far away from zero)
 gradual lines have small slopes (numbers close to zero)



horizontal line,
zero slope,
equation $\,y = k\,$

vertical line,
no slope,
equation $\,x = k\,$

UPHILL LINES, POSITIVE SLOPES 
 
gradual uphill,
small positive slope

steep uphill,
large positive slope

DOWNHILL LINES, NEGATIVE SLOPES

 
gradual downhill,
small negative slope

steep downhill,
large negative slope

