Lines are nothing more than an infinite set of points arrayed in a straight formation, but there are a number of ways to analyze them. We look at some of the main properties, formulas, and rules of lines.
Slope
The slope of a line is a measurement of how steeply the line climbs or falls as it moves from left to right. More technically, it is a line’s vertical change divided by its horizontal change, informally known as “the rise over run.” Given two points on a line, call them (x1, y1) and (x2, y2), the slope of that line can be calculated using the following formula:
The variable most often used to represent slope is m.
So, for example, the slope of a line that contains the points (–2, –4) and (6, 1) is m = (1 – (–4)) ⁄ (6 – (–2)) = 5⁄8.
Positive and Negative Slopes
You can easily determine whether the slope of a line is positive or negative just by looking at the line. If a line slopes uphill as you trace it from left to right, the slope is positive. If a line slopes downhill as you trace it from left to right, the slope is negative. You can get a sense of the magnitude of the slope of a line by looking at the line’s steepness. The steeper the line, the greater the slope will be; the flatter the line, the smaller the slope will be. Note that an extremely positive slope is larger then a moderately positive slope while an extremely negative slope is smaller then a moderately negative slope.
Look at the lines in the figure below and try to determine whether the slope of each line is negative or positive and which has the greatest slope:
Lines a and b have positive slopes, and lines c and d have negative slopes. In terms of slope magnitude, line a > b > c > d.
Special Slopes
For the Math IC, there are a few slopes you should recognize by sight. If you can recognize one of these lines and identify its slope without having to do any calculations, you will save yourself a lot of time.
- A line that is horizontal has a slope of zero. Since there is no “rise,” y2 – y1 = 0, and thus m = y2–y1 /x2–x1 = 0/x2–x1 = 0.
- A line that is vertical has an undefined slope. In this case, there is no “run,” and x2 – x1 = 0. Thus, m = y2–y1 /x2–x1 = y2–y1/ 0, and any fraction with 0 in its denominator is, by definition, undefined.
- A line that makes a 45º angle with a horizontal line has a slope of 1 or –1. This makes sense because the rise equals the run, and y2 – y1 = x2 – x1, or y2 – y1 = –(x2 – x1).
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Of the four lines pictured below, one has a slope of 0, one has a slope of 1, another has a slope of –1, and another has undefined slope. Decide which is which. |
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Line a has slope 0 because it is horizontal. Line b has slope –1 because it slopes downward at 45º as you move from left to right. Line c has slope 1 because it slopes upward at 45º as you move from left to right. Line d has undefined slope because it is vertical.
