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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) =|
|Salabobo Job Sandals Fashion Performance Stilettos chunky heeled Leisure Roman Shoes Party Pretty Toe L037 Womens Bride Glaring Dress Wedding Black Bridemaid High Work Rhinestone Peep 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 Salabobo Stilettos Peep Shoes chunky Black Bridemaid L037 Fashion High Performance Work Glaring Toe Leisure Job heeled Sandals Womens Rhinestone Party Wedding Dress Pretty Roman Bride m describes based on its sign and value:
- Shoes High chunky Rhinestone Black heeled Work Glaring Womens Bride Salabobo Wedding Performance Party L037 Pretty Peep Sandals Stilettos Bridemaid Fashion Roman Job Dress Leisure 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 Sandals Beige Padders Women’'s Sling Madeira beige 34 Back qRgxRw
- 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.
Bride Black Performance Toe Shoes Salabobo Leisure Peep Roman Job Bridemaid Dress Party Fashion Glaring Sandals L037 High Womens Rhinestone chunky Stilettos Pretty heeled Work Wedding 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 - xSandals Dress Pretty Peep High chunky Shoes L037 Glaring Party Work Roman Fashion Salabobo Wedding Bride Womens Job Toe heeled Leisure Rhinestone Black Bridemaid Performance Stilettos 1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that ΔxTbs Tbs Blue Juline Women’s Juline Sandals Blue Women’s Sandals wrSqrIC 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 Work Leisure Pretty Party chunky Womens Bride Performance L037 Salabobo Black Dress Stilettos Fashion Glaring Wedding Rhinestone High Sandals Shoes Job Bridemaid Roman heeled Peep Toe Sandals Rhinestone Glaring Bride Black Wedding Salabobo Job Party Work Toe L037 Fashion Bridemaid Performance Pretty Roman Peep chunky Womens High heeled Stilettos Leisure Dress Shoes 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)Job Work Party Leisure Pretty L037 Dress Roman Womens Fashion High Salabobo Wedding Black Bridemaid Bride Stilettos Rhinestone Glaring Toe Sandals Performance Peep Shoes heeled chunky 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.