slope

Dictionary: slope (slōp) pronunciation

v., sloped, slop·ing, slopes.

v.intr.

To diverge from the vertical or horizontal; incline: a roof that slopes. See synonyms at slant.
To move on a slant; ascend or descend: sloped down the trail.

v.tr.

To cause to slope: sloped the path down the bank.

An inclined line, surface, plane, position, or direction.
A stretch of ground forming a natural or artificial incline: ski slopes.
1. A deviation from the horizontal.
2. The amount or degree of such deviation.
Mathematics.
1. The rate at which an ordinate of a point of a line on a coordinate plane changes with respect to a change in the abscissa.
2. The tangent of the angle of inclination of a line, or the slope of the tangent line for a curve or surface.
Offensive Slang. Used as a disparaging term for a person of East Asian birth or descent.

[Probably from Middle English aslope, sloping.]

sloper slop'er n.
slopingly slop'ing·ly adv.

Search unanswered questions...

Enter a word or phrase...

5min Related Video: slope

Top

Sci-Tech Encyclopedia: Slope

The trigonometric tangent of the angle α that a line makes with the x axis. In the illustration the slope of a plane curve C at a point P of C is the slope of the line that is tangent to C at P. If y = f(x) is an equation in rectangular coordinates of curve C, the slope of C at P(x₀, y₀) is the value of the derivative dy/dx =f ′(x) at P, denoted by f ′(x₀), and hence an equation of the nonvertical tangent to C at P is y−y₀ = f ′(x₀) (x−x₀). See also Analytic geometry; Calculus.

Slope of a curve.

Computer Desktop Encyclopedia: signal slope

Top

In a chip, the time it takes for a signal to switch from 0 to 1 or 1 to 0. Although extremely fast, it is not instantaneous and can be measured in picoseconds (ps) and nanoseconds (ns).

Download Computer Desktop Encyclopedia to your iPhone/iTouch

Accounting Dictionary: Slope

Top

Tangent of the angle between a given straight line and the x-axis. It is equal to (y₂ - y₁) / (x₂ - x₁) when (x₁,y₁) and (x₂,y₂) are two distinct points on a nonvertical line. The slope indicates generally the steepness and direction of the line. More specifically, the slope is the change in y for every unit change in x. Slope is a necessary parameter for utilization of Linear Regression models. It is b in the Cost-Volume Formula y = a + bx.

Thesaurus: slope

Top

verb

See

noun

See

Antonyms: slope

Top

Definition: slant, tilt
Antonyms: evenness, level

v

Definition: slant, tilt
Antonyms: even, level

Britannica Concise Encyclopedia: slope

Top

Numerical measure of a line's inclination relative to the horizontal. In analytic geometry, the slope of any line, ray, or line segment is the ratio of the vertical to the horizontal distance between any two points on it ("slope equals rise over run"). In differential calculus, the slope of a line tangent to the graph of a function is given by that function's derivative and represents the instantaneous rate of change of the function with respect to change in the independent variable. In the graph of a position function (representing the distance traveled by an object plotted against elapsed time), the slope of a tangent line represents the object's instantaneous velocity.

For more information on slope, visit Britannica.com.

Architecture: slope

Top

1. See grade.
2. See pitch, 3.
3. See incline.
4. See grain slope.

Veterinary Dictionary: slope

Top

In statistical terms the slope of a line depicting the relationship of two variables is the gradient of the line or the regression coefficient of the relationship. A positive slope implies that increasing one variable will increase the other.

Word Tutor: slope

Top

IN BRIEF: n. - An elevated geological formation; v. - Be at an angle.

I learned to ski on the bunny slope.

Dream Symbol: Slope

Top

A slope may indicate the direction one's business or personal life is headed. An obtuse decline indicates a slow descent and a sharp incline suggests the dreamer needs to pay more attention to his or her responsibilities and also to the duties that others are obligated to perform on his or her behalf.

Wikipedia: Slope

Top

This article is about the mathematical term. For other uses, see Slope (disambiguation).

Look up slope in Wiktionary, the free dictionary.

The slope of a line is defined as the rise over the run,

m = Δ y / Δ x .

Slope is used to describe 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.

Using calculus, one can calculate the slope of the tangent to 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.

1 Definition
2 Examples
3 Geometry
- 3.1 Slope of a road or railway
4 Algebra
5 Calculus
6 See also
7 External links

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:

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

(The delta symbol, "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 x₂ - x₁, while the change in y is y₂ - y₁. Substituting both quantities into the above equation obtains the following:

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

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.

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:

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

The slope is $\textstyle\frac{1}{2} = 0.5\,$ .

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

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

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 meaning it has "no slope."

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

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

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

and

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

where angle is in degrees and the trigonometry 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.).

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

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 (x₀, y₀) on the line are both known, then the equation of the line can be found using the point-slope formula:

$y - y_0 = m(x - x_0) \,.$

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 a linear equation in the general form:

$ax + by + c = 0 \,$

is given by the formula:

$\frac {-a}{b}. \; \,$

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 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).

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.

External links

Interactive applet demonstrates how to calculate slope of a line

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Translations: Slope

Top

Dansk (Danish)
n. - skråning, hældning, skrænt, rejsning
v. intr. - skråne, gå på skrå, stå på skrå, falde
v. tr. - gøre skrå, skære skråt til

idioms:

slope arms gevær i hvil!

Nederlands (Dutch)
hellen, glooien, helling, glooiing, hellingsgraad

Français (French)
n. - (gén) pente, inclinaison, flanc, versant
v. intr. - (Mil) porter les armes
v. tr. - être en pente, pencher, descendre vers

idioms:

at the slope sur l'épaule (une arme)
slope arms (Mil) portez armes (excl)
slope off se barrer

Deutsch (German)
v. - sich neigen, abschrägen
n. - Abhang, Gefälle, Neigung

idioms:

at the slope eine Neigung haben
slope arms das Gewehr schultern
slope off plötzlich verschwinden,um eine Arbeit vermeiden zu machen

Ελληνική (Greek)
v. - γέρνω, κλίνω, ανηφορίζω, κατηφορίζω
n. - πλαγιά, πρανές, κλίση, ανωφέρεια ή κατωφέρεια

idioms:

slope arms κρατώ όπλο επ' ώμου

Italiano (Italian)
inclinare, pendio, inclinazione

idioms:

slope arms fucile in spalla, spallarm

Português (Portuguese)
v. - descer, fugir
n. - declive (m), ladeira (f)

idioms:

slope arms posicionamento da arma com suporte no ombro esquerdo quando em sentido

Русский (Russian)
уклон, склон, крутизна характеристики (элк.), наклонная выработка, положение с винтовкой на плечо, наклонный, иметь наклон, клониться, наклонить, взять на плечо (винтовку), скашивать, удрать, съехать, не уплатив за квартиру

idioms:

slope arms на плечо!

Español (Spanish)
n. - pendiente, vertiente, inclinación
v. intr. - inclinarse, tomar una dirección oblicua, estar en declive, salir de estampía, huir
v. tr. - inclinar, segar, dar una inclinación o declive

idioms:

at the slope (mil) llevar el rifle al hombro
slope arms ¡armas al hombro!
slope off largarse

Svenska (Swedish)
v. - slutta, luta, ge sig iväg, luta på, göra sluttande, snedda av, dosera (tekn)
n. - lutning, resning, dosering, sluttning, backe

中文（简体）(Chinese (Simplified))
斜坡, 倾斜, 斜面, 逃走, 使倾斜, 掮, 使有坡度

idioms:

slope arms 扛枪

中文（繁體）(Chinese (Traditional))
n. - 斜坡, 傾斜, 斜面
v. intr. - 傾斜, 逃走
v. tr. - 使傾斜, 掮, 使有坡度

idioms:

slope arms 扛槍

한국어 (Korean)
n. - 경사면, 경사도, 기울기
v. intr. - 경사지다, (죄수가) 방에서 배설물을 들어내다, 도망치다
v. tr. - ~을 경사지게 하다, (총 등을) 비스듬히 메다

日本語 (Japanese)
n. - 坂, 斜面, 傾斜
v. - 傾斜する, 坂になる

idioms:

slope arms 担え銃をする

العربيه (Arabic)
‏(فعل) ينصرف, ينحدر, يميل (الاسم) أنحدار, ميل‏

עברית (Hebrew)
n. - ‮שיפוע, מדרון, הכתפה‬
v. intr. - ‮השתפע, נטה‬
v. tr. - ‮שיפע‬

If you are unable to view some languages clearly, click here.

To select your translation preferences click here.

Best of the Web: slope

Top

Some good "slope" pages on the web:

American Sign Language
commtechlab.msu.edu

Math
mathworld.wolfram.com

Shopping: slope

Top

rocket dog ski slope

Learn More

	downgrade
	(in archaeology)
	appareille

	A truck 3500kg if drive up a slope that is 0.5km at the top of the slope your 200m higher in elevation what is the force has your engine exerted to drove up the slope? Read answer...
	If you have to do 2.2 of work to push a 78- trunk 3.1 along a slope inclined upward at 22 pushing parallel to the slope What is the coefficient of friction between trunk and slope? Read answer...
	A sled weighing 100 N is held in place on a frictionless 20 degree slope by a rope attached to a stake at the top. The rope is parallel to the slope What is the normal force of the slope acting on? Read answer...

Help us answer these

	Choose two other points from the line graph in figure 11.12 and compute the slope of the graph from these points Compare the slope that you computed with the slope between points 1 20 and 260?
	How do you find the slope intercept and standard forms of a line when the slope is undefined and you are given the coordinates 2 and 4?
	Write the equation in slope-intercept form of the line that has a slope of 2 and contains the point 1 1?

Post a question - any question - to the WikiAnswers community:

Copyrights:

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Read more
		Computer Desktop Encyclopedia. THIS COPYRIGHTED DEFINITION IS FOR PERSONAL USE ONLY. All other reproduction is strictly prohibited without permission from the publisher. © 1981-2009 Computer Language Company Inc. All rights reserved. Read more
		Accounting Dictionary. Dictionary of Accounting Terms. Copyright © 2005 by Barron's Educational Series, Inc. All rights reserved. Read more
		Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
		Antonyms. © 1999-2009 by Answers Corporation. All rights reserved. Read more
		Britannica Concise Encyclopedia. Britannica Concise Encyclopedia. © 2006 Encyclopædia Britannica, Inc. All rights reserved. Read more
		Architecture. McGraw-Hill Dictionary of Architecture and Construction. Copyright © 2003 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Word Tutor. Copyright © 2004-present by eSpindle Learning, a 501(c) nonprofit organization. All rights reserved. eSpindle provides personalized spelling and vocabulary tutoring online; free trial. Read more
		Dream Symbol. The Dreams Encyclopedia. 1995 ©Visible Ink Press. All rights reserved. Read more
		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Slope". Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

slope

Contents

Definition

Examples

Geometry

Slope of a road or railway

Algebra

Calculus

See also

External links