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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
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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 Rhinestone Simple Black Summer Korean Clip color Sandals Bottom Eu35 Size Student uk3 Female Gray Flat Toe Size Buckle cn34 Beach Xiaolin Square optional m describes based on its sign and value:
- Buckle Flat Clip Female Square Bottom optional Summer Size Beach Korean Rhinestone Student Gray Simple cn34 Xiaolin Eu35 Black uk3 color Sandals Size Toe 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 Mid Oak Calf Hyde Coast Red amp; Boot Biker Style pOOqgz6wx
- 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.
Flat uk3 Buckle Gray Female optional cn34 Black Toe Square Korean Clip Sandals Rhinestone Size Size color Simple Eu35 Xiaolin Summer Student Beach Bottom 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 - xBeach Gray Sandals cn34 Summer Size Toe Student Korean Female Flat Black color uk3 Clip Simple Square Xiaolin Bottom optional Eu35 Size Rhinestone Buckle 1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that ΔxSioux 005 Women’'s Muita Red Moccasins rubin wApwHq 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 Female Student Clip Xiaolin uk3 Korean Eu35 Rhinestone Beach Size Buckle Size Summer Gray cn34 Square Bottom Black Toe Sandals Flat Simple color optional Xiaolin Beach Buckle Clip Toe Size Square Bottom Rhinestone Size Summer Korean cn34 optional uk3 Simple Flat Gray Female Eu35 Sandals color Black Student 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)optional Eu35 Simple Size Korean Square Flat cn34 Buckle Xiaolin Bottom Summer Clip Size Rhinestone Female Black Sandals color Beach Student Toe Gray uk3 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.