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| {{About|the mathematical term}}
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| {{for|the grade (incline or gradient or pitch or slope) of any physical feature|Grade (slope)}}
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| [[File:Wiki_slope_in_2d.svg|right|thumb|Slope: <math>m=\frac{\Delta y}{\Delta x}=\tan( \theta )</math>]]
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| In [[mathematics]], the '''slope''' or '''gradient''' of a [[Line (mathematics)|line]] is a number that describes both the ''direction'' and the ''steepness'' of the line.<ref>{{cite web | url=http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |title=Oxford Concise Dictionary of Mathematics, Gradient | first1=C.|last1=Clapham|first2=J.|last2=Nicholson | publisher =Addison-Wesley | year =2009|page=348|accessdate=September 2013}}</ref> Slope is often denoted by the letter ''m''.<ref>{{cite web|last=Weisstein|first=Eric W.|title="Slope"|publisher=MathWorld--A Wolfram Web Resource|url=http://mathworld.wolfram.com/Slope.html|accessdate=September 2013}}</ref>
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| * The ''direction'' of a [[Line (mathematics)|line]] is either increasing, decreasing, horizontal or vertical.
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| **A line is '''increasing''' if it goes '''up''' towards the right. The slope is '''positive''', i.e. <math>m>0</math>.
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| **A line is '''decreasing''' if it goes '''down''' towards the right. The slope is '''negative''', i.e. <math>m<0</math>.
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| **If a line is horizontal the slope is '''zero'''. This is a [[constant function]].
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| **If a line is vertical the slope is ''undefined'' (see below).
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| * The ''steepness'', incline, or grade of a line is measured by the [[absolute value]] of the slope. A slope with a greater absolute value indicates a steeper line
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| Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.
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| The rise of a road between two points is the difference between the altitude of the road at those two points, say ''y''<sub>1</sub> and ''y''<sub>2</sub>, or in other words, the rise is (''y''<sub>2</sub> − ''y''<sub>1</sub>) = Δ''y''. For relatively short distances - where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (''x''<sub>2</sub> − ''x''<sub>1</sub>) = Δ''x''. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.
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| In mathematical language, the slope ''m'' of the line is
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| :<math>m=\frac{y_2-y_1}{x_2-x_1}.</math>
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| The concept of slope applies directly to [[grade (slope)|grade]]s or [[gradient]]s in [[geography]] and [[civil engineering]]. Through [[trigonometry]], the grade ''m'' of a road is related to its angle of incline ''θ'' by the [[Trigonometric functions#Sine, cosine and tangent|tangent function]]
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| :<math>m = \tan (\theta)\!</math>
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| Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1.
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| As a generalization of this practical description, the mathematics of [[differential calculus]] defines the slope of a [[curve]] at a point as the slope of the [[Tangent|tangent line]] at that point. When the curve given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.
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| This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.
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| == Definition ==
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| [[File:Slope of lines illustrated.jpg|thumb|400px|right|Slope illustrated for ''y'' = (3/2)''x'' − 1. Click on to enlarge]]
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| 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:
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| :<math>m = \frac{\Delta y}{\Delta x} = \frac{\text{vertical} \, \text{change} }{\text{horizontal} \, \text{change} }= \frac{\text{rise}}{\text{run}}.</math>
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| (The Greek letter ''[[delta (letter)|delta]]'', Δ, is commonly used in mathematics to mean "difference" or "change".)
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| Given two points (''x''<sub>1</sub>,''y''<sub>1</sub>) and (''x''<sub>2</sub>,''y''<sub>2</sub>), the change in ''x'' from one to the other is {{nowrap|''x''<sub>2</sub> − ''x''<sub>1</sub>}} (''run''), while the change in ''y'' is {{nowrap|''y''<sub>2</sub> − ''y''<sub>1</sub>}} (''rise''). Substituting both quantities into the above equation generates the formula:
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| :<math>m = \frac{y_2 - y_1}{x_2 - x_1}.</math>
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| The formula fails for a vertical line, parallel to the ''y'' axis (see [[Division by zero]]), where the slope can be taken as [[infinity|infinite]], so the slope of a vertical line is considered undefined.
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| === Examples ===
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| 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:
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| :<math>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}</math>.
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| :Since the slope is positive, the direction of the line is increasing. Since |m|<1, the incline is not very steep (incline <45°).
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| As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is
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| :<math>m = \frac{ 21 - 15}{3 - 4} = \frac{6}{-1} = -6.</math>
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| :Since the slope is negative, the direction of the line is decreasing. Since |m|>1, this decline is fairly steep (decline >45°).
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| ==Algebra and Geometry==
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| *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
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| ::<math>y = mx + b \,</math> | |
| :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, that is, the ''y''-coordinate where the line intersects the ''y''-axis.
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| *If the slope ''m'' of a line and a point (''x''<sub>1</sub>,''y''<sub>1</sub>) on the line are both known, then the equation of the line can be found using the [[Linear equation#Point–slope form|point-slope formula]]:
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| ::<math>y - y_1 = m(x - x_1).\!</math>
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| *The slope of the line defined by the [[linear equation]]
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| ::<math>ax + by +c = 0 \,</math>
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| :is
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| ::<math>-\frac {a}{b} \;</math>.
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| *Two lines are [[parallel (geometry)|parallel]] if and only if their slopes are equal and they are not the same line (coincident) or if they both are vertical and therefore both have undefined slopes. Two lines are [[perpendicular]] 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).
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| *The angle θ between -90° and 90° that a line makes with the ''x''-axis is related to the slope ''m'' as follows:
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| ::<math>m = \tan (\theta)</math>
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| :and
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| ::<math>\theta = \arctan (m)</math> (this is the inverse function of tangent; see [[trigonometry]]).
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| ===Examples===
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| For example, consider a line running through the points (2,8) and (3,20). This line has a slope, ''m'', of
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| ::<math>\frac {(20 - 8)}{(3 - 2)} \; = 12. \,</math>
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| :One can then write the line's equation, in point-slope form:
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| ::<math>y - 8 = 12(x - 2) = 12x - 24 \,</math>
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| :or:
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| ::<math>y = 12x - 16. \,</math>
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| :The angle θ between -90° and 90° that this line makes with the ''x'' axis is
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| ::<math>\theta=\arctan (12) \approx 85.2^{\circ} \,.</math>
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| Consider the two lines: ''y'' = -3''x'' + 1 and ''y'' = -3 ''x'' - 2. Both lines have slope ''m'' = -3. They are not the same line. So they are parallel lines.
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| Consider the two lines ''y'' = -3''x'' + 1 and ''y'' = <sup>''x''</sup>/<sub>3</sub> - 2. The slope of the first line is ''m''<sub>1</sub> = -3. The slope of the second line is ''m''<sub>2</sub> = <sup>1</sup>/<sub>3</sub>. The product of these two slopes is -1. So these two lines are perpendicular.
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| == Slope of a road or railway ==
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| :''Main articles: [[Grade (slope)]], [[Grade separation]]''
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| There are two common ways to describe the steepness of a [[road]] or [[Rail tracks|railroad]]. One is by the angle between 0° and 90° (in degrees), and the other is by the slope in a percentage. See also [[steep grade railway]] and [[rack railway]].
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| The formulae for converting a slope given as a percentage into an angle in degrees and vice versa are:
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| ::<math>\text{angle} = \arctan \left( \frac{\text{slope}}{100%} \right)</math> , (this is the inverse function of tangent; see [[trigonometry]])
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| :and
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| ::<math>\mbox{slope} = 100% \cdot \tan( \mbox{angle}),\, </math>
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| where ''angle'' is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100[[Percent sign|%]] or 1000[[Per mil|‰]] is an angle of 45°.
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| 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 ''"1 in 10"'', ''"1 in 20"'' etc.) Note that 1:10 is steeper than 1:20. For example, steepness of 20% means 1:5 or an incline with angle 11,3°.
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| <gallery>
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| File:Nederlands verkeersbord J6.svg|Slope warning sign in the [[Netherlands]]
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| File:Znak A-23.svg|Slope warning sign in [[Poland]]
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| File:Skloník-klesání.jpg|A 1371-meter distance of a railroad with a 20[[Per mil|‰]] slope. [[Czech Republic]]
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| File:Railway gradient post.jpg|Steam-age railway gradient post indicating a slope in both directions at [[Meols railway station]], [[United Kingdom]]
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| </gallery>
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| ==Calculus==
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| [[File:Tangent function animation.gif|right|frame|At each point, the [[derivative]] is the slope of a [[Line (geometry)|line]] that is [[tangent]] to the [[curve]] at that point. Note: the derivative at the point A is [[positive number|positive]] where green and dash-dot, [[negative number|negative]] where red and dashed, and [[zero (number)|zero]] where black and solid.]]
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| 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.
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| 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,
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| :<math>m = \frac{\Delta y}{\Delta x}</math>,
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| 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.
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| For example, the slope of the secant intersecting ''y'' = ''x''<sup>2</sup> at (0,0) and (3,9) is 3. (The slope of the tangent at {{nowrap|1=x = {{frac|3|2}}}} is also 3—''a'' consequence of the [[mean value theorem]].)
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| 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|limit]], 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, or ''dy''/''dx''. We call this limit the [[derivative (calculus)|derivative]].
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| :<math>\frac{dy}{dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}</math>
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| Its value at a point on the function gives us the slope of the tangent at that point. For example, let ''y''=''x''<sup>2</sup>. A point on this function is (-2,4). The derivative of this function is <sup>d''y''</sup>/<sub>d''x''</sub>=2''x''. So the slope of the line tangent to ''y'' at (-2,4) is 2·(-2) = -4. The equation of this tangent line is: ''y''-4=(-4)(''x''-(-2)) or ''y'' = -4''x'' - 4.
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| ==Other generalizations==
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| The concept of slope can be generalized to functions of more than one variable where it is more often referred to as [[gradient]].
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| ==References==
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| {{reflist}}
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| ==See also==
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| * [[Linear function (calculus)|Linear function]]
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| * [[Trigonometric function#Slope definitions|Slope definitions]]
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| * [[Euclidean distance]]
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| * [[Inclined plane]]
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| * [[Theil–Sen estimator]], a line with the [[median]] slope among a set of sample points
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| == External links ==
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| {{cite web | url=http://www.mathopenref.com/coordslope.html| title =Slope of a Line (Coordinate Geometry)| publisher =Math Open Reference|year=2009|accessdate=September 2013}} interactive
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| {{Wiktionary|slope}}
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| [[Category:Elementary mathematics]]
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| [[Category:Analytic geometry]]
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