In mathematics, the slope (or gradient, especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of said line. This pages focuses on such slopes. With an understanding of algebra and geometry, one can calculate the slope of a straight line; with calculus, one can calculate the slope of a curve at a point.

The concept of slope, and much of this article, applies directly to grades or gradients in geography and civil engineering.

Definition of slope

It is generally represented by m, and 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:

and memorized as "rise over run" or change in y over change in x. (The triangular symbol "Δ" is the Greek letter delta, commonly used in mathematics to mean "change". So m is equal to the change in y, the vertical coordinate, divided by the change in x, the horizontal coordinate; that is m is the ratio of the changes.) This concept is fundamental to algebra, analytic geometry, trigonometry, and calculus.

Note that it doesn't matter which two points on the line you pick, or in which order you use them: the same line will always have the same slope. Curves have "accelerating" slopes and one can use calculus to determine such slopes.

Example 1

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

The slope is 1/2 = 0.5.

Example 2

If a line runs through the points (4, 15) and (3, 21) then:

Geometry

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:

and

(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.

Slope of a road, etc.

There are two common ways to describe how steep a road, etc., is: by the angle in degrees or by the slope in a percentage.

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:

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

Calculus

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 said point, and is thus equal to the rate of change of the function at that point.

Why calculus is necessary

The slope given by m = Δy / Δx (where Δy and Δx are the distances along both axes between two points on a curve) is the slope of a secant line to the curve. With a line, the secant between any two points is equivalent to the line/curve itself; therefore, the above formula gives the slope of the line. With a non-linear curve, this is not the case.

For example, the average slope of y = x², from x = 0 to x = 3, is m = 9 / 3 = 3 (which happens to be the actual slope at, and only at, x = 1.5).

However, by moving the points used in the above formula closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve. The secant line is only equivalent to the tangent line when Δy and Δx equal zero; however, this obtains a slope of 0/0, which is undefined (division by zero). The concept of a limit is necessary to calculate this slope; the slope is the limit of Δy / Δx as Δy and Δx approach zero. This is more conveniently written as dy/dx.