Inmathematics, theslope orgradient of aline is a number that describes thedirection of the line on aplane.[1] Often denoted by the letterm, slope is calculated as theratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same A slope is the ratio of the vertical distance (rise) to the horizontal distance (run) between two points, not a direct distance or a direct angle for any choice of points. To explain, a slope is theratio of the vertical distance (rise) to thehorizontal distance (run) between two points, not a directdistance or a directangle. The line may be physical – as set by aroad surveyor, pictorial as in adiagram of a road or roof, orabstract.An application of the mathematical concept is found in thegrade orgradient ingeography andcivil engineering.
Thesteepness, incline, or grade of a line is theabsolute value of its slope: greater absolute value indicates a steeper line. The line trend is defined as follows:
An "increasing" or "ascending" line goesup from left to right and has positive slope:.
A "decreasing" or "descending" line goesdown from left to right and has negative slope:.
A "horizontal" line (the graph of aconstant function) has zero slope:.
A "vertical" line has undefined or infinite slope (see below).
If two points of a road have altitudesy1 andy2, the rise is the difference (y2 −y1) = Δy. Neglecting theEarth's curvature, if the two points have horizontal distancex1 andx2 from a fixed point, the run is (x2 −x1) = Δx. The slope between the two points is thedifference ratio:
Thus, a 45° rising line has slopem = +1, and a 45° falling line has slopem = −1.
Generalizing this,differential calculus defines the slope of aplane curve at a point as the slope of itstangent line at that point. When the curve is approximated by a series of points, the slope of the curve may be approximated by the slope of thesecant line between two nearby points. When the curve is given as the graph of analgebraic expression, calculus givesformulas for the slope at each point. Slope is thus one of the central ideas of calculus and its applications to design.
There seems to be no clear answer as to why the letterm is used for slope, but it first appears in English inO'Brien (1844)[2] who introduced the equation of a line as"y =mx +b", and it can also be found inTodhunter (1888)[3] who wrote "y =mx +c".[4]
Slope illustrated fory = (3/2)x − 1. Click on to enlargeSlope of a line in coordinates system, fromf(x) = −12x + 2 tof(x) = 12x + 2
The slope of a line in the plane containing thex andy axes is generally represented by the letterm,[5] and is defined as the change in they coordinate divided by the corresponding change in thex coordinate, between two distinct points on the line. This is described by the following equation:
(The Greek letterdelta, Δ, is commonly used in mathematics to mean "difference" or "change".)
Given two points and, the change in from one to the other is (run), while the change in is (rise). Substituting both quantities into the above equation generates the formula:
The formula fails for a vertical line, parallel to the axis (seeDivision by zero), where the slope can be taken asinfinite, so the slope of a vertical line is considered undefined.
Suppose a line runs through two points:P = (1, 2) andQ = (13, 8). By dividing the difference in-coordinates by the difference in-coordinates, one can obtain the slope of the line:
Since the slope is positive, the direction of the line is increasing. Since |m| < 1, the incline is not very steep (incline < 45°).
As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is
Since the slope is negative, the direction of the line is decreasing. Since |m| > 1, this decline is fairly steep (decline > 45°).
If is alinear function of, then the coefficient of is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form
then is the slope. This form of a line's equation is called theslope-intercept form, because can be interpreted as they-intercept of the line, that is, the-coordinate where the line intersects the-axis.
If the slope of a line and a point on the line are both known, then the equation of the line can be found using thepoint-slope formula:
Two lines areparallel if and only if they are not the same line (coincident) and either their slopes are equal or they both are vertical and therefore both have undefined slopes.
Two lines areperpendicular 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).
The angle θ between −90° and 90° that a line makes with thex-axis is related to the slopem as follows:
For example, consider a line running through points (2,8) and (3,20). This line has a slope,m, of
One can then write the line's equation, in point-slope form:
or:
The angle θ between −90° and 90° that this line makes with thex-axis is
Consider the two lines:y = −3x + 1 andy = −3x − 2. Both lines have slopem = −3. They are not the same line. So they are parallel lines.
Consider the two linesy = −3x + 1 andy =x/3 − 2. The slope of the first line ism1 = −3. The slope of the second line ism2 =1/3. The product of these two slopes is −1. So these two lines are perpendicular.
At each point, thederivative is the slope of aline that istangent to thecurve at that point. Note: the derivative at point A ispositive where green and dash–dot,negative where red and dashed, andzero where black and solid.
The concept of a slope is central todifferential calculus. For non-linear functions, the rate of change varies along the curve. Thederivative of the function at a point is the slope of the linetangent 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 thex andy axes, respectively) between two points on a curve, then the slope given by the above definition,
,
is the slope of asecant 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 intersectingy =x2 at (0,0) and (3,9) is 3. (The slope of the tangent atx =3⁄2 is also 3 − a consequence of themean 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. Usingdifferential calculus, we can determine thelimit, or the value that Δy/Δx approaches as Δy and Δx get closer tozero; it follows that this limit is the exact slope of the tangent. Ify is dependent onx, 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, or dy/dx. We call this limit thederivative.
The value of the derivative at a specific point on the function provides us with the slope of the tangent at that precise location. For example, lety =x2. A point on this function is (−2,4). The derivative of this function isdy⁄dx = 2x. So the slope of the line tangent toy at (−2,4) is2 ⋅ (−2) = −4. The equation of this tangent line is:y − 4 = (−4)(x − (−2)) ory = −4x − 4.
The illusion of a paradox of area is dispelled by comparing slopes where blue and red triangles meet.
An extension of the idea of angle follows from the difference of slopes. Consider theshear mapping
Then is mapped to. The slope of is zero and the slope of is. The shear mapping added a slope of. For two points on with slopes and, the image
has slope increased by, but the difference of slopes is the same before and after the shear. This invariance of slope differences makes slope an angularinvariant measure, on a par with circular angle (invariant under rotation) and hyperbolic angle, with invariance group ofsqueeze mappings.[7][8]
The slope of a roof, traditionally and commonly called theroof pitch, in carpentry and architecture in the US is commonly described in terms of integer fractions of one foot (geometric tangent, rise over run), a legacy of British imperial measure. Other units are in use in other locales, with similar conventions. For details, seeroof pitch.
There are two common ways to describe the steepness of aroad orrailroad. One is by the angle between 0° and 90° (in degrees), and the other is by the slope in a percentage. See alsosteep grade railway andrack railway.
The formulae for converting a slope given as a percentage into an angle in degrees and vice versa are:
(this is the inverse function of tangent; seetrigonometry)
and
whereangle is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100% or 1000‰ is an angle of 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 (or "1in 10", "1in 20", etc.) 1:10 is steeper than 1:20. For example, steepness of 20% means 1:5 or an incline with angle 11.3°.
Roads and railways have both longitudinal slopes and cross slopes.
The concept of a slope or gradient is also used as a basis for developing other applications in mathematics:
Gradient descent, a first-order iterative optimization algorithm for finding the minimum of a function
Gradient theorem, theorem that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve
Gradient method, an algorithm to solve problems with search directions defined by the gradient of the function at the current point
Conjugate gradient method, an algorithm for the numerical solution of particular systems of linear equations
^O'Brien, M. (1844),A Treatise on Plane Co-Ordinate Geometry or the Application of the Method of Co-Ordinates in the Solution of Problems in Plane Geometry, Cambridge, England: Deightons
^Todhunter, I. (1888),Treatise on Plane Co-Ordinate Geometry as Applied to the Straight Line and Conic Sections, London: Macmillan
^Weisstein, Eric W."Slope". MathWorld--A Wolfram Web Resource.Archived from the original on 6 December 2016. Retrieved30 October 2016.
