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[parent] slope (Definition)

The slope of a straight line in the $ xy$-plane expresses how great is the change of the ordinate $ y$ of the point of the line per a unit-change of the abscissa $ x$ of the point; it requires that the line is not vertical.

The slope $ m$ of the line may be determined by taking the changes of the coordinates between two arbitrary points $ (x_1, y_1)$ and $ (x_2, y_2)$ of the line:

$\displaystyle m = \frac{y_2-y_1}{x_2-x_1}$

The equation of the line is

$\displaystyle y = mx+b,$
where $ b$ indicates the intersection point of the line and the $ y$-axis (one speaks of y-intercept).

The slope is equal to the tangent of the slope angle of the line.

Two non-vertical lines of the plane are parallel if and only if their slopes are equal.

\includegraphics{slope}

In the previous picture, the blue line given by $ 3x-y+1 = 0$ has slope $ 3$, whereas the red one given by $ 2x+y+2 = 0$ has slope $ -2$. Also notice that positive slopes represent ascending graphs and negative slopes represent descending graphs.



"slope" is owned by pahio. [ full author list (2) ]
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See Also: derivative, example of rotation matrix, parallellism in Euclidean plane, slope angle, line in plane, difference quotient

Other names:  angle coefficient (?)
Keywords:  parallelism of lines

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Attachments:
proof that the slope of a line is well-defined (Proof) by Dr_Absentius
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Cross-references: negative, graphs, positive, parallel, plane, slope angle, y-intercept, intersection, equation, coordinates, abscissa, point, ordinate, line
There are 44 references to this entry.

This is version 11 of slope, born on 2004-11-08, modified 2008-02-16.
Object id is 6457, canonical name is Slope.
Accessed 6066 times total.

Classification:
AMS MSC: 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
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What is "suuntakulma" in English? by pahio on 2007-06-05 11:51:24
Hi, has the angle which a line in the xy-plane forms with the positive x-axis some own name? I have used in a pair of entries the name "direction angle", which is a direct translation from Finnish. (The Germans speak of "Steigungswinkel" or "Neigungswinkel.)
Jussi
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