AskDefine | Define slope

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 five-degree gradient" [syn: gradient] v : be at an angle; "The terrain sloped down" [syn: incline, pitch]

User Contributed Dictionary

English

Pronunciation

Noun

  1. An area of ground that tends evenly upward or downward.
    I had to climb a small slope to get to the site.
  2. The degree to which a surface tends upward or downward.
    The road has a very sharp downward slope at that point.
  3. 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
  4. The slope of the line tangent to a curve at a given point.
    The slope of a parabola increases linearly with x.
  5. (vulgar, highly offensive) A person of Chinese or other East Asian descent.

Synonyms

Translations

area of ground that tends evenly upward or downward
degree to which a surface tends upward or downward
mathematics
offensive: a person of East Asian descent

Verb

  1. To tend steadily upward or downward.
    The road slopes sharply down at that point.
  2. (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

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.
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.

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.
(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.
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.
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 = \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
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 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
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:
\mbox = \arctan \frac ,
and
\mbox = 100 \tan( \mbox),\,
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.).

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 \; = 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 \; \,.

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.
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

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
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