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.

- Definition
- Examples
- Algebra and geometry
- Examples 2
- Statistics
- Slope of a road or railway
- Calculus
- Difference of slopes
- Other uses
- See also
- References
- External links

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**: . - A
*decreasing*line goes*down*from left to right and has**negative slope**: . - A
*horizontal*line (the graph of a constant function) has**zero slope**:**.** - A
*vertical*line has undefined or**infinite slope**(see below).

If two points of a road have altitudes *y*_{1} and *y*_{2}, the rise is the difference (*y*_{2} − *y*_{1}) = Δ*y*. Neglecting the Earth's curvature, if the two points have horizontal distance *x*_{1} and *x*_{2} from a fixed point, the run is (*x*_{2} − *x*_{1}) = Δ*x*. The slope between the two points is the **difference ratio**:

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

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] }

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:

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

Given two points and , the change in from one to the other is (*run*), while the change in is (*rise*). Substituting both quantities into the above equation generates the formula:

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

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

- 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

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

- If is a linear function of , then the coefficient of is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form
*slope-intercept form*, because can be interpreted as the y-intercept of the line, that is, the -coordinate where the line intersects the -axis. - If the slope of a line and a point on the line are both known, then the equation of the line can be found using the point-slope formula:
- The slope of the line defined by the linear equation
is

- .

- Two lines are parallel if and only if they are not the same line (coincident) and either their slopes are equal or they both are vertical and therefore both 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).
- The angle θ between −90° and 90° that a line makes with the
*x*-axis is related to the slope*m*as follows:and

- (this is the inverse function of tangent; see inverse trigonometric functions).

For example, consider a line running through 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 angle θ between −90° and 90° that this line makes with the *x*-axis is

Consider the two lines: *y* = −3*x* + 1 and *y* = −3*x* − 2. Both lines have slope *m* = −3. They are not the same line. So they are parallel lines.

Consider the two lines *y* = −3*x* + 1 and *y* = *x*/3 − 2. The slope of the first line is *m*_{1} = −3. The slope of the second line is *m*_{2} = 1/3. The product of these two slopes is −1. So these two lines are perpendicular.

In statistics, the gradient of the least-squares regression best-fitting line for a given sample of data may be written as:

- ,

This quantity *m* is called as the * regression slope * for the line . The quantity is Pearson's correlation coefficient, is the standard deviation of the y-values and is the standard deviation of the x-values. This may also be written as a ratio of covariances:^{ [6] }

There are two common ways to describe the steepness of a road or railroad. One is by the angle between 0° and 90° (in degrees), and the other is by the slope in a percentage. See also steep grade railway and rack railway.

The formulae for converting a slope given as a percentage into an angle in degrees and vice versa are:

- (this is the inverse function of tangent; see trigonometry)

and

where *angle* is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100% or 1000‰ is an angle of 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 (or "1 *in* 10", "1 *in* 20", etc.) 1:10 is steeper than 1:20. For example, steepness of 20% means 1:5 or an incline with angle 11.3°.

Roads and railways have both longitudinal slopes and cross slopes.

- Slope warning sign in the Netherlands
- Slope warning sign in Poland
- A 1371-meter distance of a railroad with a 20‰ slope. Czech Republic
- Steam-age railway gradient post indicating a slope in both directions at Meols railway station, United Kingdom

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 Δ*x* and Δ*y* be the distances (along the *x* and *y* axes, respectively) between two points on a curve, then the slope given by the above definition,

- ,

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* = *x*^{2} at (0,0) and (3,9) is 3. (The slope of the tangent at *x* = 3⁄2 is also 3 − *a* consequence of the mean value theorem.)

By moving the two points closer together so that Δ*y* and Δ*x* 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 Δ*y*/Δ*x* approaches as Δ*y* and Δ*x* 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 Δ*x* approaches zero. Therefore, the slope of the tangent is the limit of Δ*y*/Δ*x* as Δ*x* approaches zero, or d*y*/d*x*. We call this limit the derivative.

The value of the derivative at a specific point on the function provides us with the slope of the tangent at that precise location. For example, let *y* = *x*^{2}. A point on this function is (−2,4). The derivative of this function is d*y*⁄d*x* = 2*x*. So the slope of the line tangent to *y* at (−2,4) is 2 ⋅ (−2) = −4. The equation of this tangent line is: *y* − 4 = (−4)(*x* − (−2)) or *y* = −4*x* − 4.

An extension of the idea of angle follows from the difference of slopes. Consider the shear mapping

Then is mapped to . The slope of is zero and the slope of is . The shear mapping added a slope of . For two points on with slopes and , the image

has slope increased by , but the difference of slopes is the same before and after the shear. This invariance of slope differences makes slope an angular invariant measure, on a par with circular angle (invariant under rotation) and hyperbolic angle, with invariance group of squeeze mappings.^{ [7] }^{ [8] }

The concept of a slope or gradient is also used as a basis for developing other applications in mathematics:

- Gradient descent, a first-order iterative optimization algorithm for finding the minimum of a function
- Gradient theorem, theorem that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve
- Gradient method, an algorithm to solve problems with search directions defined by the gradient of the function at the current point
- Conjugate gradient method, an algorithm for the numerical solution of particular systems of linear equations
- Nonlinear conjugate gradient method, generalizes the conjugate gradient method to nonlinear optimization
- Stochastic gradient descent, iterative method for optimizing a differentiable objective function

- Euclidean distance
- Grade
- Inclined plane
- Linear function
- Line of greatest slope
- Mediant
- Slope definitions
- Theil–Sen estimator, a line with the median slope among a set of sample points

In mathematics, **analytic geometry**, also known as **coordinate geometry** or **Cartesian geometry**, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

A **centripetal force** is a force that makes a body follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

In mathematics, a **hyperbola** is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

In mathematics, a **parabola** is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, the **trigonometric functions** are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

In geometry, the **tangent line** (or simply **tangent**) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is tangent to the curve *y* = *f*(*x*) at a point *x* = *c* if the line passes through the point (*c*, *f*(*c*)) on the curve and has slope *f*'(*c*), where *f*' is the derivative of *f*. A similar definition applies to space curves and curves in *n*-dimensional Euclidean space.

In mathematics, **differential calculus** is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

The **grade** (US) or **gradient** (UK) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction in which *run* is the horizontal distance and *rise* is the vertical distance.

In mathematics, the **inverse trigonometric functions** are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

The **great-circle distance**, **orthodromic distance**, or **spherical distance** is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path between the two points on the surface of the sphere.

In mathematical physics, **scalar potential** describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.

In geometry, an **envelope** of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

The **external secant** function is a trigonometric function defined in terms of the secant function:

An **osculating circle** is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. The osculating circle provides a way to understand the local behavior of a curve and is commonly used in differential geometry and calculus.

**Arc length** is the distance between two points along a section of a curve.

In mathematics and computational science, **Heun's method** may refer to the **improved** or **modified Euler's method**, or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods.

The **differentiation of trigonometric functions** is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′(*a*) = cos(*a*), meaning that the rate of change of sin(*x*) at a particular angle *x = a* is given by the cosine of that angle.

In geometry, the **tangential angle** of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis.

In Euclidean plane geometry, a **tangent line to a circle** is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.

- ↑ Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Gradient" (PDF). Addison-Wesley. p. 348. Archived from the original (PDF) on 29 October 2013. Retrieved 1 September 2013.
- ↑ O'Brien, M. (1844),
*A Treatise on Plane Co-Ordinate Geometry or the Application of the Method of Co-Ordinates in the Solution of Problems in Plane Geometry*, Cambridge, England: Deightons - ↑ Todhunter, I. (1888),
*Treatise on Plane Co-Ordinate Geometry as Applied to the Straight Line and Conic Sections*, London: Macmillan - ↑ Weisstein, Eric W. "Slope". MathWorld--A Wolfram Web Resource. Archived from the original on 6 December 2016. Retrieved 30 October 2016.
- ↑ An early example of this convention can be found in Salmon, George (1850).
*A Treatise on Conic Sections*(2nd ed.). Dublin: Hodges and Smith. pp. 14–15. - ↑
*Further Mathematics Units 3&4 VCE (Revised)*. Cambridge Senior Mathematics. 2016. ISBN 9781316616222 – via Physical Copy. - ↑ Bolt, Michael; Ferdinands, Timothy; Kavlie, Landon (2009). "The most general planar transformations that map parabolas into parabolas".
*Involve: A Journal of Mathematics*.**2**(1): 79–88. doi: 10.2140/involve.2009.2.79 . ISSN 1944-4176. Archived from the original on 2020-06-12. Retrieved 2021-05-22. - ↑ Abstract Algebra/Shear and Slope at Wikibooks

Look up ** slope ** in Wiktionary, the free dictionary.

- "Slope of a Line (Coordinate Geometry)". Math Open Reference. 2009. Retrieved 30 October 2016. interactive

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