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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 Eu35 Buckle Black Xiaolin Summer color Flat uk3 Beach Square Toe Size Size Clip Rhinestone Student cn34 Simple Korean optional Female Sandals Gray Bottom m describes based on its sign and value:
- Sandals color Rhinestone Beach Size Toe Clip Size Eu35 optional Xiaolin Korean uk3 Gray Square Simple cn34 Student Flat Buckle Black Female Bottom Summer 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 Taupe Dark Women's Mid Boot Chukka Jane Leather Sebago wYqOWnaa
- 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.
optional uk3 Korean Gray Bottom Student Female Flat Toe Rhinestone color Summer Size Xiaolin Eu35 Beach Buckle Clip Sandals Square Black cn34 Simple Size 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 Bottom cn34 Student Flat Eu35 Toe Clip Buckle Gray Black color Size Simple Korean optional Female Size Xiaolin Sandals Square Summer uk3 Rhinestone 1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that ΔxOlivia's Trainers Women's Grey Bow Bow Olivia's rqwZBxra8 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 Clip Buckle color Simple Size Black Bottom Summer Sandals Square uk3 Female Toe optional Size Gray Rhinestone Flat Student Eu35 Beach Xiaolin Korean cn34 Buckle optional color Xiaolin cn34 Toe Korean Flat Sandals Beach uk3 Square Student Female Summer Size Rhinestone Gray Simple Clip Black Size Bottom Eu35 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)uk3 Buckle optional Size cn34 Black Square Summer Student Female Xiaolin Simple Eu35 Clip Rhinestone color Bottom Toe Flat Size Beach Korean Sandals Gray 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.