Slope
Term used in mathematics.
The slope of a line is defined as "rise over run," which means the change in y over the change in x.
In the commonly used slope-intercept form for a line, y = mx + b, the slope is indicated by the constant m.
You can determine the slope between any two points in the Cartesian coordinate system by dividing the difference of the y values by the difference in x values. For example, given points P(1,2) and Q(13,8), we can find the slope m.
y1 - y2
m = ————————
x1 - x2
So,
8 - 2 6 1
PQm = ——————— = ——— = ———
13 -1 12 2
and we found that the slope is 1/2. Note that it is also equivalent if we switch the points:
2 - 8 -6 1
PQm = ——————— = ——— = ———
1 -13 -12 2
Given the slope of a line (y=mx+b) and the y intercept</y> (where x=0), we can determine the equation of the line.
.
In our example with points P(1,2), Q(13,8) and slope 1/2, the y intercept</y> is 3/2. So using our slope-intercept form for a line we have, y = 1/2x + 3/2. b is the y intercept, hence the name of this form, slope-intercept.
(Note: In this example we gave you the y intercept of 3/2, which you could have derived using the point-slope form of an equation for a line.)
Often it is easier to just remember an equation for the slope of a line called "point-slope" form:
y - yo = m (x - xo)
Given any point (xo, yo) and a slope m, this form will give the equation for the line described by them.
A horizontal line has a slope of 0 (the "rise" being 0).
The slope of a vertical line is undefined. The problem with attempting to use limits to define the slope as infinity is that such techniques could produce a slope of either positive or negative infinity.
The angle the line makes with respect to the x axis can be used to calculate the slope. m = tan θ. (See trigonometry).
The concept of the slope of a line is fundamental to algebra, analytic geometry, trigonometry, and calculus to name a few.