Slope
In mathematics, the slope of a straigth line in a Cartesian coordinate system is a measure for the "steepness" of the line. It is defined as the change in y divided by the corresponding change in x (if the horizontal axis is the x-axis and the vertical axis is the y-axis), often written as
- m = Δy / Δx
and memorized as "rise over run". The concept of the slope of a line is fundamental to algebra, analytic geometry, trigonometry, and calculus.
Concretely, to compute the slope of a given line, pick any two points on the line, say P(13,8) and Q(1,2), and divide the difference in y-coordinates by the difference in x-coordinates:
Δy y1 - y2 8 - 2 6 1
m = —— = ———————— = ——————— = ——— = ——
Δx x1 - x2 13 - 1 12 2
and we found that the slope is 1/2. Note that it doesn't matter which two points on the line you pick or in which order you use them: you will always get the same slope.
Geometric meaning
The larger the slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. The slope of a vertical line is not defined (it does not make sense to define it as +∞, because it might just as well be defined as -∞).
The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:
- m = tan θ
and
- θ = arctan m
(see trigonometry).
Two lines are parallel if and only if their slopes are equal; they are perpendicular (i.e. they form a right angle) if and only if the product of their slopes is -1.
Algebra
If the equation of the line is given in the form
- y = mx + b
then the slope m can be read off as the coefficient of the x variable. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis.
If you know the slope m of a line and a point (x0, y0) on the line, then you can find the equation of the line using the point-slope formula:
- y - y0 = m (x - x0)
Calculus
The concept of slope is also central to differential calculus. The slope is only defined for straight lines, while calculus deals with more complicated curved functions. To find the slope or steepness of such a function at a given point, one first finds the tangent to the graph of the function at that point. This tangent is a straight line which touches the graph of the function. The slope of the tangent can then be computed; it is the steepness of the function at that point, and it is called the derivative of the function at that point. Unlike in the case of straight lines, the derivative of curved functions changes from point to point.