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]
- Rhymes: -əʊp
- 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.
area of ground that tends evenly upward or downward
degree to which a surface tends upward or downward
offensive: a person of East Asian descent
to tend steadily upward or downward
to try to move surreptitiously
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.
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 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.
ExamplesSuppose 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.
GeometryThe 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 roadroad 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 ,
- \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.).
AlgebraIf 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 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 \,.
- 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 \; \,.
CalculusThe 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.
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
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