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 m describes based on its sign and value:
Slope is essentially the change in height over the 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 - x1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that Δx 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 Δx and Δy form a right triangle, it is possible to calculate d using the Pythagorean theorem. Refer to the Triangle Calculator 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)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 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.  The slope calculator determines the slope or gradient between two points in the Cartesian coordinate system. The slope is basically the amount of slant a line has and can have a positive, negative, zero, or undefined value. Before using the calculator, it is probably worth learning how to find the slope using the slope formula. To find the equation of a line for any given two points that this line passes through, use our slope intercept form calculator.
To find the slope of a line we need two coordinates on the line. Any two coordinates will suffice. We are basically measuring the amount of change of the y-coordinate, often known as the rise, divided by the change of the x-coordinate, known the the run. The calculations in finding the slope are simple and involves nothing more than basic subtraction and division.
slope = (y₂ - y₁) / (x₂ - x₁) Notice that the slope of a line is easily calculated by hand using small, whole number coordinates. The formula becomes increasingly useful as the coordinates take on larger values or decimal values. It is worth mentioning that any horizontal line has a gradient of zero because a horizontal line has the same y-coordinates. This will result in a zero in the numerator of the slope formula. On the other hand, a vertical line will have an undefined slope since the x-coordinates will always be the same. This will result the division by zero error when using the formula.
Just as slope can be calculated using the endpoints of a segment, the midpoint can also be calculated. The midpoint is an important concept in geometry, particularly when inscribing a polygon inside another polygon with the its vertices touching the midpoint of the sides of the larger polygon. This can be obtained using the midpoint calculator or by simply taking the average of each x-coordinates and the average of the y-coordinates to form a new coordinate. The slopes of lines are important in determining whether or not a triangle is a right triangle. If any two sides of a triangle have slopes that multiply to equal -1, then the triangle is a right triangle. The computations for this can be done by hand or by using the right triangle calculator. You can also use the distance calculator to compute which side of a triangle is the longest, which helps determine which sides must form a right angle if the triangle is right. The sign in front of the gradient provided by the slope calculator indicates whether the line is increasing, decreasing, constant or undefined. If the graph of the line moves from lower left to upper right it is increasing and is therefore positive. If it decreases when moving from the upper left to lower right, then the gradient is negative.
The method for finding the slope from an equation will vary depending on the form of the equation in front of you. If the form of the equation is y=mx+c, then the slope (or gradient) is just m. If the equation is not in this form, try to rearrange the equation. To find the gradient of other polynomials, you will need to differentiate the function with respect to x.
A 1/20 slope is one that rises by 1 unit for every 20 units traversed horizontally. So, for example, a ramp that was 200 ft long and 10 ft tall would have a 1/20 slope. A 1/20 slope is equivalent to a gradient of 1/20 (strangely enough) and forms an angle of 2.86° between itself and the x-axis.
As the slope of a curve changes at each point, you can find the slope of a curve by differentiating the equation with respect to x and, in the resulting equation, substituting x for the point at which you’d like to find the gradient.
The rate of change of a graph is also its slope, which are also the same as gradient. Rate of change can be found by dividing the change in the y (vertical) direction by the change in the x (horizontal) direction, if both numbers are in the same units, of course. Rate of change is particularly useful if you want to predict the future of previous value of something, as, by changing the x variable, the corresponding y value will be present (and vice versa).
Slopes (or gradients) have a number of uses in everyday life. There are some obvious physical examples - every hill has a slope, and the steeper the hill, the greater its gradient. This can be useful if you are looking at a map and want to find the best hill to cycle down. You also probably sleep under a slope, a roof that is. The slope of a roof will change depending on the style and where you live. But, more importantly, if you ever want to know how something changes with time, you will end up plotting a graph with a slope.
A 10% slope is one that rises by 1 unit for every 10 units travelled horizontally (10 %). For example, a roof with a 10% slope that is 20 m across will be 2 m high. This is the same as a gradient of 1/10, and an angle of 5.71° is formed between the line and the x-axis.
To find the area under a slope you need to integrate the equation and subtract the lower bound of the area from the upper bound. For linear equations:
A 5 to 1 slope is one that, for every increase of 5 units horizontally, rises by 1 unit. The number of degrees between a 5 to 1 slope and the x-axis is 11.3°. This can be found by first calculating the slope, by dividing the change in the y direction by the change in the x direction, and then finding the inverse tangent of the slope. |
What is the slope of the line that passes through these two points (8, 4) and (5, 3)
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