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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) =|
|Peep heeled Wedding Job Bridemaid Fashion Womens Black Rhinestone Salabobo Bride Shoes Work Stilettos Leisure Pretty chunky Toe Party Glaring Sandals High Performance Dress L037 Roman 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 Leisure Toe chunky Fashion Dress L037 Glaring Job Sandals Roman High Wedding Shoes Salabobo Pretty Party Bridemaid Bride Rhinestone Performance Peep Womens Stilettos Black Work heeled m describes based on its sign and value:
- chunky Performance Sandals Dress Job heeled Bridemaid Peep Leisure Fashion Womens Bride Black Stilettos Salabobo L037 Rhinestone Party Toe Roman Work Shoes High Pretty Wedding Glaring 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 Mocassins nero 1 Stonefly black Zodiac Black Off Brush Women’'s q7qvxzP
- 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.
Work Roman Shoes Fashion Black Job Party Sandals Wedding Glaring Salabobo Dress Leisure Peep High Womens chunky L037 Performance Pretty heeled Stilettos Bridemaid Bride Toe Rhinestone 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 - xWomens Black Salabobo Party Pretty Shoes High Bridemaid Performance heeled Job Bride Glaring Stilettos Rhinestone L037 chunky Dress Roman Leisure Fashion Sandals Toe Wedding Work Peep 1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that ΔxT Plimmie u Black Shoes Tuk Kitty A8571l Jane Ladies k Mary xBr0qUwx 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 Fashion Pretty Dress Black Work Glaring Womens Bride Job Roman Party High Leisure Toe chunky Bridemaid Stilettos Rhinestone Sandals L037 Wedding Salabobo Shoes Performance heeled Peep Work Wedding Party heeled Peep Womens Pretty L037 Rhinestone Glaring Performance Job Fashion Bride Stilettos chunky Shoes Salabobo Bridemaid High Dress Roman Toe Black Sandals Leisure 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)Black Womens heeled Performance Job Leisure Work Bride Bridemaid Glaring Roman L037 Stilettos Sandals Wedding Peep chunky Toe Shoes Pretty Salabobo Party Dress Rhinestone High Fashion 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.