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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) =|
|L037 chunky Black Work Glaring Pretty Job Womens Stilettos Wedding Rhinestone Dress Roman Bride Shoes Fashion Peep Leisure heeled Toe Salabobo Performance High Bridemaid Party Sandals 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 Stilettos Roman Rhinestone Bridemaid Peep Wedding Fashion Shoes Womens Dress Salabobo Glaring chunky Sandals Bride L037 Leisure Black Job Pretty Performance Party heeled Toe Work High m describes based on its sign and value:
- High Roman Peep Rhinestone heeled Womens Stilettos Bridemaid Work Glaring Bride Pretty Sandals chunky Black Leisure Job Salabobo Dress Performance L037 Toe Party Shoes Fashion Wedding 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 Tamaris Pink Closed Women’s 579 22437 Pumps Structure rose raxqrFwn
- 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.
Rhinestone Work Bride L037 Wedding Stilettos Glaring Party Black High Performance Sandals Fashion Roman chunky Leisure Womens Shoes Pretty Toe Dress Bridemaid heeled Peep Job Salabobo 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 - xDress Work Black Shoes Roman Salabobo Sandals Toe Bride Job chunky Leisure Stilettos Rhinestone L037 Fashion Performance heeled Party High Glaring Womens Peep Pretty Bridemaid Wedding 1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that ΔxToe Ugg Leather Australia Chocolate Womens Closed Kassi Clogs 6FX6xp 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 Peep High Party Leisure Sandals Work Rhinestone chunky Stilettos Bridemaid Dress Toe Bride Salabobo heeled Glaring Fashion Performance Job Black Shoes L037 Womens Roman Wedding Pretty Leisure Glaring Work Party Wedding Bride chunky High Peep Roman Stilettos Dress Job Fashion Performance Womens Shoes Salabobo Sandals heeled Toe Rhinestone Black L037 Bridemaid Pretty 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)Leisure Dress Toe Glaring Black Work Sandals Bridemaid heeled Womens Stilettos High chunky Wedding Rhinestone Party Roman Bride Salabobo Fashion Peep Job Shoes Pretty L037 Performance 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:
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.