Information AboutSlope |
| CATEGORIES ABOUT SLOPE | |
| elementary mathematics | |
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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 Grade s or Gradient s 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: : (The ''delta'' Symbol , " Δ ", is commonly used in mathematics to mean "difference" or "change".) Given two points (''x''1, ''y''1) and (''x''2, ''y''2), the change in ''x'' from one to the other is ''x''2 - ''x''1, while the change in ''y'' is ''y''2 - ''y''1. Substituting both quantities into the above equation obtains the following: : 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 Curve s 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: : 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 : 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 : : and : (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 or railroad 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: : and : . 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.). |
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