In mathematics, the slope (or gradient, especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of said line. This pages focuses on such slopes. With an understanding of algebra and geometry, one can calculate the slope of a straight line; with calculus, one can calculate the slope of a curved line at a point.
The concept of slope, and much of this article, applies directly to gradess or gradients in geography and civil engineering.
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2 Geometry 3 Algebra 4 Calculus |
Note that it doesn't matter which two points on the line you pick, or in which order you use them: the same line will always have the same slope. Curves have "accelerating" slopes and one can use calculus to determine such slopes.
If a line runs through the points (4, 15) and (3, 21) then:
The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:
Two lines are parallel if and only if their slopes are equal; they are perpendicular (i.e. they form a right angle) if and only if the product of their slopes is -1.
If the equation of the line is given in the form
then the slope m can be read off as the coefficient of the x variable. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis.If you know the slope m of a line and a point (x_{0}, y_{0}) on the line, then you can find the equation of the line using the point-slope formula:
For example, the average slope of y = x², from x = 0 to x = 3, is m = 9 / 3 = 3 (which happens to be the actual slope at, and only at, x = 1.5). Should one attempt to use the above formula for a single point, such as (x = 3, y = 9); one then gets m = 0 / 0 ; as the &Delta for both y and x equals zero (see also: division by zero).
The concept of a slope is central to differential calculus; which deals with functions whose graph is not a line. Unlike linear functions, the slope of a non-linear function varies at different points. This slope is often referred to as a derivative. To find the slope at a given point on a curve, one must find a line which is tangent to said function, at said point. The slope of said tangential line is equal to the slope of said function at said point.