Slope Calculator

Understanding Slope in Mathematics

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. 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:

• 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.
• A vertical line has an undefined slope since it would result in division by zero.

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, provided the distance is not large enough that Earth’s curvature becomes relevant.

Slope Formula

m = (y₂ – y₁) / (x₂ – x₁)

In this equation:

• Δy = y₂ – y₁ (vertical change)
• Δx = x₂ – x₁ (horizontal change)

Distance Between Two Points

d = √((x₂ – x₁)² + (y₂ – y₁)²)

Angle of Incline

m = tan(θ)

Given two points (3, 4) and (6, 8):

Finding Slope:

m = (8 – 4) / (6 – 3) = 4 / 3

Finding Distance:

d = √((6 – 3)² + (8 – 4)²) = 5

Finding Angle:

4 / 3 = tan(θ)

θ = tan^-1(4 / 3) ≈ 53.13°

While this is beyond the scope of this calculator, aside from its basic linear use, the concept of 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 represented by the slope of the line tangent to the curve at that point.

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