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Slope is often used to describe the measurement of the steepness, incline, gradient, or grade (slope) of a line (mathematics). 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 grade (slope)s or gradients in
geography and civil engineering.
Definition of slope
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 math 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}.
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 Δ
y is the "rise" and Δ
x 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 "
acceleration" 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
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.
The angle θ a line makes with the positive
x axis is closely related to the slope
m via the tangent
function (mathematics):
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 Rail tracks 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{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% 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.).Image:Ten percent slope.svg|slope warning sign,
NetherlandsImage:Znak_A-23.svg]Image:Skloník-klesání.jpg|A 1371 meters distance of a railroad with a 20
‰ slope.
Czech RepublicImage:Railway gradient post.jpg], 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
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,
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 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 of a function, 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. We call this limit the derivative (calculus).
See also
Slope is often used to describe the measurement of the steepness, incline, gradient, or grade (slope) of a line (mathematics). 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
grade (slope)s or
gradients in
geography and
civil engineering.
Definition of slope
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 math 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}.
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 Δ
y is the "rise" and Δ
x 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 "
acceleration" 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
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.
The angle θ a line makes with the positive
x axis is closely related to the slope
m via the tangent function (mathematics):
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
Rail tracks 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{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% 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.).Image:Ten percent slope.svg|slope warning sign,
NetherlandsImage:Znak_A-23.svg]Image:Skloník-klesání.jpg|A 1371 meters distance of a railroad with a 20
‰ slope.
Czech RepublicImage:Railway gradient post.jpg],
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
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,
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 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 of a function, 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. We call this limit the derivative (calculus).
See also
Slope - Wikipedia, the free encyclopedia
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 ...
Definition: slope from Online Medical Dictionary
The Online Medical Dictionary is a searchable dictionary of definitions from medicine, science and technology.
JNLL Bracknell Dry Ski & Snowboard Centre - Home
Ski Slope Times and Prices. Ice Rink Timetable. Ice Rink Activity Prices. Ice Rink Course Prices ... Welcome to the John Nike Leisuresport Complex
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Learn to ski and snowboard on our breath-taking indoor skiing and snowboarding slope at Chill Factore, Manchester. Great family days out at our Alpine Village with real snow
JNLL Chatham Dry Ski & Snowboard Centre - Home
The Centre has the longest artificial slope in the South East and is the premier wintersports facility in the region. It is also part of the Company which teaches more people to ...
JNLL Plymouth Dry Ski & Snowboard Centre - Home
With three slopes to choose from including the longest artificial slope in the South West, Plymouth Ski and Snowboard Centre is the perfect place to practice and learn new skills.
Slope: New poetry, criticism and commentary
Online journal of original poetry. Current and past issues. Part of Slope Publishing Inc., a non-profit organization registered in the State of New Hampshire.
Learn to Ski, Learn to Snowboard at the Sheffield Ski Village ...
An all year dry ski slope ski resort with Snow Mountain, Adventure Mountain, Urban Jungle, Village Inn and Snow and Rock.
Ski slope facilities
Gloucester Ski & Snowboard Centre offers a wealth of facilities including a 220 metre main dry ski slope, a 150m trainer slope, a Fun Park, 2 nursery slopes and 3 drag lifts.
BSSOA - Slope Finder
The British Ski Slope Operators Association. Ski slope facilities, and contact information for the UK ... Slope Telephone Web site Photo; Aberdeen Snowsports Centre : Alford Ski ...
