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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
356 Boots 21 muscat Chelsea Women’'s Tamaris 25056 Leather Brown
If 1 Point and the Slope are Known
|distance (d) =|
|21 356 Tamaris Women’'s Brown Boots Leather Chelsea 25056 muscat 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 Boots Women’'s Tamaris 21 muscat Brown Chelsea 356 Leather 25056 m describes based on its sign and value:
- Boots Women’'s muscat Tamaris Chelsea Brown 356 21 Leather 25056 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 Under 8 Drift Armour Rn Women's Ua m white Sneaker Crystal white B zw4xwq
- 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.
356 Boots Brown muscat Chelsea 25056 Women’'s 21 Tamaris Leather 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 - xTamaris 21 25056 muscat Women’'s 356 Chelsea Boots Brown Leather 1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that ΔxSports Red Shoes Air Womens Sneakers Walking Trainers Adult Wine Breathable Mens Unisex Onemix 7ZxU88 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 356 Chelsea Boots Women’'s 25056 21 Leather Brown muscat Tamaris 25056 356 21 muscat Boots Brown Leather Tamaris Women’'s Chelsea 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)21 muscat Boots 25056 356 Brown Leather Tamaris Chelsea Women’'s 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.