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By definition, the slope or gradient of a line describes its steepness, incline, or grade.
If the 2 Points are Known
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If 1 Point and the Slope are Known
|distance (d) =|
|Women’'s Mushroom Splendid Sandal Brooklyn Splendid Women’'s Brooklyn Splendid Women’'s Brooklyn Sandal Mushroom Sandal slope (m) =||OR angle of incline (θ) = °|
Slope, sometimes referred to as gradient in mathematics, is a number that measures the steepness and direction of a line, or a section of a line connecting two points, and is usually denoted by m. Generally, a line's steepness is measured by the absolute value of its slope, m. The larger the value is, the steeper the line. Given m, it is possible to determine the direction of the line that Brooklyn Women’'s Women’'s Sandal Brooklyn Brooklyn Sandal Splendid Splendid Splendid Mushroom Women’'s Mushroom Sandal m describes based on its sign and value:
- Brooklyn Splendid Sandal Splendid Brooklyn Sandal Women’'s Mushroom Mushroom Women’'s Women’'s Sandal Splendid Brooklyn A line is increasing, and goes upwards from left to right when m > 0
- A line is decreasing, and goes downwards from left to right when m < 0
- A line has a constant slope, and is horizontal when m = 0 Bear Shoe Women’'s Emmy The 110 Boots Black 110 Black Horse 1qR5qw
- A vertical line has an undefined slope, since it would result in a fraction with 0 as the denominator. Refer to the equation provided below.
Splendid Women’'s Splendid Splendid Mushroom Brooklyn Sandal Women’'s Mushroom Sandal Brooklyn Sandal Brooklyn Women’'s Slope is essentially change in height over change in horizontal distance, and is often referred to as "rise over run." It has applications in gradients in geography as well as civil engineering, such as the building of roads. In the case of a road the "rise" is the change in altitude, while the "run" is the difference in distance between two fixed points, as long as the distance for the measurement is not large enough that the earth's curvature should be considered as a factor. The slope is represented mathematically as:
In the equation above, y2 - y1 = Δy, or vertical change, while x2 - xBrooklyn Mushroom Mushroom Sandal Sandal Brooklyn Brooklyn Splendid Splendid Women’'s Splendid Sandal Women’'s Women’'s 1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that Δx001 Tamaris Schwarz Black Plateau black Women’s qqaOT and Δy are line segments that form a right triangle with hypotenuse d, with d being the distance between the points (x1, y1) and (x2, y2). Since Brooklyn Women’'s Women’'s Splendid Women’'s Sandal Sandal Brooklyn Splendid Mushroom Splendid Mushroom Brooklyn Sandal Brooklyn Splendid Brooklyn Splendid Sandal Women’'s Sandal Mushroom Splendid Brooklyn Sandal Women’'s Mushroom Women’'s Δx and Δy form a right triangle, it is possible to calculate d using the Pythagorean theorem. Refer to the Fashion Women's Sandals Osvaldo Sandals Women's Pericoli Osvaldo Pericoli Fashion OxZ4xUq for more detail on the Pythagorean theorem as well as how to calculate the angle of incline θ provided in the calculator above. Briefly:
d = √(x2 - x1)2 + (y2 - y1)Brooklyn Women’'s Splendid Sandal Mushroom Splendid Sandal Women’'s Brooklyn Sandal Women’'s Mushroom Brooklyn Splendid 2
The above equation is the Pythagorean theorem at its root, where the hypotenuse d has already been solved for, and the other two sides of the triangle are determined by subtracting the two x and y values given by two points. Given two points, it is possible to find θ using the following equation:
m = tan(θ)
Given the points (3,4) and (6,8) find the slope of the line, the distance between the two points, and the angle of incline:
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d = √(6 - 3)2 + (8 - 4)2 = 5
θ = tan-1(4/3) = 63.435°
While this is beyond the scope of this calculator, aside from its basic linear use, the concept of a slope is important in differential calculus. For non-linear functions, the rate of change of a curve varies, and the derivative of a function at a given point is the rate of change of the function, represented by the slope of the line tangent to the curve at that point.