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Slope

Slope


Slope is a quantity that measures the steepness of a line. To figure out how to define slope, think about what it means for one line to be steeper than another. Intuitively, one would say the steeper line "climbs faster." To make this mathematically precise, consider the following. If two points on the steeper line that are horizontally one unit apart are compared with two points on the less-steep line that are horizontally one unit apart, the pair on the steeper line will be farther apart vertically. Thus the slope of a line is defined to be the ratio of the vertical distance to the horizontal distance between any two points on that line. Sometimes this ratio is called "rise to run," meaning the vertical change relative to the horizontal change.

Several interesting things can be noticed about the slope of a line. First of all, the definition does not tell us which two points on the line to choose. Fortunately, this does not matter because any pair of points on the same line will yield the same slope.

In fact, this is really the defining characteristic of a line: When a line is "straight," this means that its steepness never varies, unlike a parabola or a circle, which "bend" and therefore climb more steeply in some places than in others.

The slope of a line indicates whether the line slants upwards from left to right, slants downwards from left to right, or is flat. If a line slants upwards, movement from one point to a second point is such that one unit to the right of the first will yield an increase (that is, a positive difference) in the y -coordinates of the points. If the line slants downward, the points' y -coordinates will decrease (a negative difference) as one unit is moved to the right. If the line is horizontal, the y -coordinate will not change. Therefore, an upward-slanting line has positive slope, a downward-slanting line has negative slope, and a horizontal line has a slope of zero.

A slope also involves division, which raises the possibility of dividing by zero. This would occur whenever the slope of a vertical line is computed (and only then). Therefore, vertical lines are considered to have an undefined slope. Alternatively, some people consider the slope of a vertical line to be infinite, which makes sense intuitively because a vertical line climbs as steeply as possible.

see also Graphs, Effects of Parameter Changes; Infinity; Lines, Parallel and Perpendicular; Lines, Skew.

Naomi Klarreich

Bibliography

Charles, Randall I., et al. Focus on Algebra. Menlo Park, CA: Addison-Wesley, 1996.

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slope

slope / slōp/ • n. 1. a surface of which one end or side is at a higher level than another; a rising or falling surface: he slithered helplessly down the slope. ∎  a difference in level or sideways position between the two ends or sides of a thing: the roof should have a slope sufficient for proper drainage | the backward slope of the chair. ∎  (often slopes) a part of the side of a hill or mountain, esp. as a place for skiing: a ten-minute cable-car ride delivers you to the slopes. ∎  the gradient of a graph at any point. ∎  Electr. the transconductance of a valve, numerically equal to the gradient of one of the characteristic curves of the valve. 2. inf., offens. an Asian person, esp. a Vietnamese or other Southeast Asian. • v. [intr.] (of a surface or line) be inclined from a horizontal or vertical line; slant up or down: the garden sloped down to a stream the ceiling sloped | [as adj.] (sloping) a sloping floor. ∎  [tr.] place or arrange in such a position or inclination: Poole sloped his shoulders | [as adj.] (sloped) a sloped leather writing surface.

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slope

slope1 †adv. in an oblique direction or position XV. Aphetic of aslope (of uncert. orig.).
Hence as vb. intr. take an oblique direction XVI, tr. (spec. mil.) bring into a slanting position XVII; and as sb. XVII.

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slope

slope2 (sl.) make off, decamp. XIX. orig. U.S.; perh. spec. use of prec. vb.

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slope

slopeaslope, cope, dope, elope, grope, hope, interlope, lope, mope, nope, ope, pope, rope, scope, slope, soap, taupe, tope, trope •myope • telescope • periscope •stereoscope • bioscope • stroboscope •kaleidoscope • CinemaScope •gyroscope • microscope • horoscope •stethoscope • antelope • envelope •zoetrope • skipping-rope • tightrope •towrope • heliotrope • lycanthrope •philanthrope • thaumatrope •misanthrope •isotope, radioisotope

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