Dictionary Definition
slope
Noun
1 an elevated geological formation; "he climbed
the steep slope"; "the house was built on the side of the mountain"
[syn: incline, side]
2 the property possessed by a line or surface
that departs from the horizontal; "a fivedegree gradient" [syn:
gradient] v : be at an
angle; "The terrain sloped down" [syn: incline, pitch]
User Contributed Dictionary
English
Pronunciation

 Rhymes: əʊp
Noun
 An area of ground that tends evenly upward or downward.
 I had to climb a small slope to get to the site.
 The degree to which a surface tends upward or downward.
 The road has a very sharp downward slope at that point.
 The ratio of the
vertical and horizontal distances between
two points on a line;
zero if the line is horizontal, infinite if it is vertical.
 The slope of this line is 0.5
 The slope of the line tangent to a curve at a given point.
 The slope of a parabola increases linearly with x.
 (vulgar, highly offensive) A person of Chinese or other East Asian descent.
Synonyms
 (area of ground that tends evenly upward or downward): bank, embankment, gradient, hill, incline
 (degree to which a surface tends upward or downward): gradient
 (mathematics): first derivative, gradient
 (offensive: Chinese person): Chinaman, Chink
Translations
area of ground that tends evenly upward or
downward
degree to which a surface tends upward or
downward
mathematics
 Czech: sklon
 Dutch: richtingscoëfficient, helling
 Finnish: kulmakerroin
 French: dérivée première
 German: Steigung
 Italian: inclinazione
 Slovene: strmina
offensive: a person of East Asian descent
 Dutch: spleetoog
 Italian: muso giallo
Verb
 To tend steadily upward or downward.
 The road slopes sharply down at that point.
 (colloquial, usually followed by a preposition) To try to move
surreptitiously.
 I sloped in through the back door, hoping my boss wouldn't see me.
Translations
to tend steadily upward or downward
to try to move surreptitiously
 Finnish: livahtaa, hiipi�
 Italian: muoversi furtivamente
Derived terms
References
Extensive Definition
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.
The concept of slope, and much of this article,
applies directly to grades or
gradients in geography and civil
engineering.
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.
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.
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 yaxis is
vertical and the xaxis 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.
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 ycoordinates by the difference in xcoordinates, one can obtain the slope of the line: m = \frac = \frac = \frac = \frac = \frac.
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
 m = \frac = \frac = 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
 \theta = \arctan\,m
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
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 undefined slope (a
vertical line).
Slope of a road
 Main articles: Grade (slope), Grade separation
 \mbox = \arctan \frac ,
 \mbox = 100 \tan( \mbox),\,
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.).
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 \,
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
pointslope 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 \; = 12 \,.
 y  8 = 12(x  2) = 12x  24 \,
 y = 12x  16 \,.
The slope of a linear
equation in the general form:
 Ax + By + C = 0 \,
 \frac \; \,.
Calculus
The concept of a slope is central to differential calculus. For nonlinear 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,
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.
See also
 The gradient is a generalization of the concept of slope for functions of more than one variable.
 Slope definitions
slope in Arabic: ميل
slope in Bulgarian: Диференчно частно
slope in Catalan: Pendent (matemàtiques)
slope in Czech: Směrnice
slope in Danish: Hældningstal
slope in German: Steigung
slope in Spanish: Pendiente de la recta
slope in French: Pente (mathématiques)
slope in Icelandic: Hallatala
slope in Italian: Coefficiente angolare
slope in Dutch: Hellingsgraad
slope in Norwegian: Stigningstall
slope in Portuguese: Talude
slope in Finnish: Kulmakerroin
slope in Swedish: Riktningskoefficient
slope in Tamil: சாய்வு
slope in Chinese: 斜率
Synonyms, Antonyms and Related Words
acclivity, angle, angularity, ascend, ascent, bank, bend, bevel, bezel, camber, cant, careen, chute, climb, decline, declivity, deflection, descend, descent, deviation, dip, downgrade, drop, drop off, easy slope,
fall, fall away, fall off,
fleam, gentle slope,
glacis, go downhill, go
uphill, grade, gradient, hanging gardens,
heel, helicline, hill, hillside, inclination, incline, inclined plane,
keel, launching ramp,
lean, leaning, leaning tower, list, mount, obliqueness, obliquity, pitch, rake, ramp, recline, retreat, rise, scarp, shelve, shelving beach, side, sidle, sink, skew, slant, steep slope, stiff climb,
swag, sway, talus, tilt, tip, tower of Pisa, upgrade, uprise