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.
Graphs, Effects of Parameter Changes; Infinity; Lines, Parallel and Perpendicular; Lines, Skew.
Bibliography
Charles, Randall I., et al. Focus on Algebra. Menlo Park, CA: Addison-Wesley, 1996.
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