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The slope of a line is defined as the rise over the run, m = Δy / Δx.

Slope is often used to describe the measurement of the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. It is also always the same thing as how many rises in one run. Look up slope in Wiktionary, the free dictionary. ... Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ... Image File history File links Slope_picture. ... Image File history File links Slope_picture. ... A grade (or gradient) is the pitch of a slope, and is often expressed as a percent tangent, or rise over run. It is used to express the steepness of slope on a hill, stream, roof, railroad, or road, where zero indicates level (with respect to gravity) and increasing numbers... Line redirects here. ...

Using calculus, one can calculate the slope of the tangent to a curve at a point. For other uses, see Calculus (disambiguation). ... For other uses, see tangent (disambiguation). ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

The concept of slope, and much of this article, applies directly to grades or gradients in geography and civil engineering. A grade (or gradient) is the pitch of a slope, and is often expressed as a percent tangent, or rise over run. It is used to express the steepness of slope on a hill, stream, roof, railroad, or road, where zero indicates level (with respect to gravity) and increasing numbers... For other uses, see Gradient (disambiguation). ... The Falkirk Wheel in Scotland. ...

Definition

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:

(The delta symbol, "Δ", is commonly used in mathematics to mean "difference" or "change".) Delta (upper case Δ, lower case δ) is the fourth letter of the Greek alphabet. ...

Given two points (x₁, y₁) and (x₂, y₂), the change in x from one to the other is x₂ - x₁, while the change in y is y₂ - y₁. Substituting both quantities into the above equation obtains the following:

Scientific Definition: The rate at which an object accelerates on a distance versus time graph is shown. Calculated by Slope = Rise / Run of a graph. Since the y-axis is vertical and the x-axis is horizontal by convention, the above equation is often memorized as "rise over run", where Δy is the "rise" and Δx is the "run". Therefore, by convention, 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. This article is about the branch of mathematics. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... For other uses, see Calculus (disambiguation). ...

Note that the way the points are chosen on the line and their order does not matter; the slope will be the same in each case. Other curves have "accelerating" slopes and one can use calculus to determine such slopes. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... For other uses, see Calculus (disambiguation). ...

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:

The slope is 1/2 = 0.5.

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

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 infinite.

The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function: Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

and

(see trigonometry). Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with...

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 inifinite slopes. Two lines are perpendicular if and only if the product of their slopes is -1 or one has a slope of 0 (a horizontal line) and the other has an infinite slope (a vertical line). Fig. ...

Slope of a road

Main articles: Grade (slope), Grade separation

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. The formulae for converting a slope as a percentage into an angle in degrees and vice versa are: A grade (or gradient) is the pitch of a slope, and is often expressed as a percent tangent, or rise over run. It is used to express the steepness of slope on a hill, stream, roof, railroad, or road, where zero indicates level (with respect to gravity) and increasing numbers... An example of a four-level stack interchange in the Netherlands. ... For other uses, see Road (disambiguation). ... Rail tracks. ... A mountain railway is a railway that ascends and descends a mountain slope that has a steep grade. ...

and

where angle is in degrees and the trigonometry functions operate in degrees. For example, a 100% 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.).

slope warning sign, Netherlands Image File history File links Ten_percent_slope. ...

slope warning sign, Poland Image File history File links This is a lossless scalable vector image. ...

A 1371-meter distance of a railroad with a 20‰ slope. Czech Republic Image File history File links Metadata Size of this preview: 800 × 600 pixelsFull resolution (2048 × 1536 pixel, file size: 684 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... A permille or per mille is a tenth of a percent or one part per thousand. ...

Steam-age railway gradient post indicating a slope in both directions at Meols railway station, United Kingdom Image File history File links Metadata Size of this preview: 800 × 450 pixelsFull resolution (3072 × 1728 pixel, file size: 2. ... British Railways London Midland Region totem sign for Meols station. ...

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 A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ...

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. The y-intercept in 2-dimensional space is the point where the graph of a function or relationship intercepts the y-axis of the coordinate system. ...

If the slope m of a line and a point (x₀, y₀) on the line are both known, then the equation of the line can be found using the point-slope formula: Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ...

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

.

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

or:

.

The slope of a linear equation in the general form: Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ...

is given by the formula:

.

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 the point, and is thus equal to the rate of change of the function at that point. Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... This article is about derivatives and differentiation in mathematical calculus. ... For other uses, see tangent (disambiguation). ...

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, Image File history File links Download high resolution version (824x560, 9 KB) , made with Graph 2. ...

,

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. A secant line of a curve is a line that intersects two or more points on the curve. ...

For example, the slope of the secant intersecting y = x² at (0,0) and (3,9) is m = (9 - 0) / (3 - 0) = 3 (which happens to be the slope of the tangent at, and only at, x = 1.5, a consequence of the mean value theorem). In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...

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 Δy/Δx 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 Δy/Δx as Δx approaches zero. We call this limit the derivative. Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... In mathematics, the limit of a function is a fundamental concept in analysis. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

See also

• The gradient is a generalization of the concept of slope for functions of more than one variable.
• Slope definitions