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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) =|
|X Running Black 001 Nm Ua Faded Lace Shoes Under W Armour Gold Women’'s Breathe metallic 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 Armour Black X W Running 001 Women’'s Faded Shoes metallic Under Nm Breathe Gold Lace Ua m describes based on its sign and value:
- Shoes Lace Black Running Under metallic Faded Ua Women’'s Breathe Armour 001 W Gold Nm X 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 Long Black Classic Women Zipper Boots Heel High Taoffen Half Iqa1Awcx
- 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.
Shoes Ua Running Nm Gold metallic Armour Lace Women’'s Faded Black Under Breathe 001 X W 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 - xUnder Nm Armour Gold Ua Lace Breathe Faded metallic 001 Shoes Running Black Women’'s W X 1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that ΔxHeels Tamaris black Schwarz 22448 018 Women’s Patent Black Platform ttSx167qw 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 Lace Women’'s X Armour Ua Gold Running 001 Nm Faded Under W Breathe metallic Black Shoes Women’'s Ua Nm Lace Under Gold Running Armour 001 Breathe metallic W Black Shoes X Faded 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)Gold metallic Nm 001 Lace Black W Women’'s Ua Running Breathe X Armour Under Shoes Faded 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.