# Slope

The slope of a line in the plane is defined as the rise over the run, m = Δyx.

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline. Slope is normally described by the ratio of the "rise" divided by the "run" between two points on a line. The line may be practical - as set by a road surveyor : or in a diagram that models a road or a roof either as a description or as a plan.

The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2y1) = Δy. For relatively short distances - where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line. In mathematical language, the slope m of the line is

$m=\frac{y_2-y_1}{x_2-x_1}.$

The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the grade m of a road is related to its angle of incline θ by

$m = \tan \theta\!$ (from the definition of tangent).

As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.

This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.

## Definition

Slope illustrated for y = (3/2)x − 1. Click on to enlarge

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation:

$m = \frac{\Delta y}{\Delta x} = \frac{\text{rise}}{\text{run}}.$

(The Greek letter delta, Δ, is commonly used in mathematics to mean "difference" or "change".)

Given two points (x1,y1) and (x2,y2), the change in x from one to the other is x2x1 (run), while the change in y is y2y1 (rise). Substituting both quantities into the above equation generates the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}.$

The formula fails for a vertical line, parallel to the y axis (see Division by zero), where the slope can be taken as infinite, so the slope of a vertical line is considered undefined.

### Examples

Suppose a line runs through two points: P = (1, 2) and Q = (13, 8). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{13 - 1} = \frac{6}{12} = \frac{1}{2}.$

As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is

$m = \frac{ 21 - 15}{3 - 4} = \frac{6}{-1} = -6.$

## Geometry

The larger the absolute value of a 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. A vertical line's slope is undefined.

The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:

$m = \tan\,\theta$

and

$\theta = \arctan\,m$

(see trigonometry).

Two lines are parallel if and only if their slopes are equal and they are not coincident or if they both are vertical and therefore have undefined slopes. Two lines are perpendicular if the product of their slopes is −1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line). Also, another way to determine a perpendicular line is to find the slope of one line and then to get its reciprocal and then reversing its positive or negative sign (e.g. a line perpendicular to a line of slope  −2 is +1/2).

## Slope of a road or railway

There are two common ways to describe how steep a road or railroad is. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway and rack railway. The formulae for converting a slope as a percentage into an angle in degrees and vice versa are:

$\text{angle} = \arctan \frac{\text{slope}}{100} ,$

and

$\mbox{slope} = 100 \tan( \mbox{angle}),\,$

where angle is in degrees and the trigonometric functions operate in degrees. For example, a 100% or 1000 slope is 45°.

A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (etc.).

## Algebra

If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form

$y = mx + b \,$

then m is the slope. 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 the slope m of a line and a point (x1,y1) on the line are both known, then the equation of the line can be found using the point-slope formula:

$y - y_1 = m(x - x_1).\!$

For example, consider a line running through the points (2,8) and (3,20). This line has a slope, m, of

$\frac {(20 - 8)}{(3 - 2)} \; = 12. \,$

One can then write the line's equation, in point-slope form:

$y - 8 = 12(x - 2) = 12x - 24 \,$

or:

$y = 12x - 16. \,$

The slope of the line defined by the linear equation

$ax + by +c = 0 \,$

is: $-\frac {a}{b} \;$.

## Calculus

At each point, the derivative is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black

The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point.

If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition,

$m = \frac{\Delta y}{\Delta x}$,

is the slope of a secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve.

For example, the slope of the secant intersecting y = x2 at (0,0) and (3,9) is 3. (The slope of the tangent at x = 32 is also 3—a consequence of the mean value theorem.)

By moving the two points closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus, we can determine the limit, or the value that Δyx approaches as Δy and Δx get closer to zero; it follows that this limit is the exact slope of the tangent. If y is dependent on x, then it is sufficient to take the limit where only Δx approaches zero. Therefore, the slope of the tangent is the limit of Δyx as Δx approaches zero, or dy/dx. We call this limit the derivative.

$\frac{dy}{dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}$