In mathematics, the slope or gradient of a line is a number that describes the direction and steepness of the line.^[1] Often denoted by the letter m, slope is calculated as the ratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same number for any choice of points. A line descending left-to-right has negative rise and negative slope. The line may be physical – as set by a road surveyor, pictorial as in a diagram of a road or roof, or abstract.

The steepness, incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. Direction is defined as follows:

• An increasing line goes up from left to right, and has positive slope: ${\displaystyle m>0}$.
• A decreasing line goes down from left to right, and has negative slope: ${\displaystyle m<0}$.
• A horizontal line (the graph of a constant function) has zero slope: ${\displaystyle m=0}$.
• A vertical line has undefined or infinite slope (see below).

If two points of a road have altitudes y₁ and y₂, the rise is the difference (y₂ − y₁) = Δy. Neglecting the Earth's curvature, if the two points have horizontal distance x₁ and x₂ from a fixed point, the run is (x₂ − x₁) = Δx. The slope between the two points is the difference ratio:

${\displaystyle m={\frac {\Delta y}{\Delta x}}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}$

This is equivalent to the grade or gradient in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of inclination θ by the tangent function

${\displaystyle m=\tan(\theta ).}$

Thus, a 45° rising line has slope m = +1, and a 45° falling line has slope m = −1.

Generalizing this, differential calculus defines the slope of a curve at a point as the slope of its tangent line at that point. When the curve is approximated by a series of points, the slope of the curve may be approximated by the slope of the secant line between two nearby points. When the curve is given as the graph of an algebraic expression, calculus gives formulas for the slope at each point. Slope is thus one of the central ideas of calculus and its applications to design.

There seems to be no clear answer as to why the letter m is used for slope, but it first appears in English in O'Brien (1844)^[2] who introduced the equation of a line as "y = mx + b", and it can also be found in Todhunter (1888)^[3] who wrote "y = mx + c".^[4]

Definition

The slope of a line in the plane containing the x and y axes is generally represented by the letter m,^[5] 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:

${\displaystyle m={\frac {\Delta y}{\Delta x}}={\frac {{\text{vertical}}\,{\text{change}}}{{\text{horizontal}}\,{\text{change}}}}={\frac {\text{rise}}{\text{run}}}.}$

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

Given two points ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$, the change in ${\displaystyle x}$ from one to the other is ${\displaystyle x_{2}-x_{1}}$ (run), while the change in ${\displaystyle y}$ is ${\displaystyle y_{2}-y_{1}}$ (rise). Substituting both quantities into the above equation generates the formula:

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}$

The formula fails for a vertical line, parallel to the ${\displaystyle y}$ 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 ${\displaystyle y}$-coordinates by the difference in ${\displaystyle x}$-coordinates, one can obtain the slope of the line:

${\displaystyle m={\frac {\Delta y}{\Delta x}}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {8-2}{13-1}}={\frac {6}{12}}={\frac {1}{2}}.}$

Since the slope is positive, the direction of the line is increasing. Since |m| < 1, the incline is not very steep (incline < 45°).

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

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

Since the slope is negative, the direction of the line is decreasing. Since |m| > 1, this decline is fairly steep (decline > 45°).

Algebra and geometry

Examples

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

${\displaystyle \frac{(20-<!--\; -\; -->8)}{(3-<!--\; -\; -->2)}}$Zdroj:https://en.wikipedia.org?pojem=Slopes
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