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Slope

What is Slope?

Slope is a measure of the steepness of a line. It tells us how much the y-value changes for each unit change in the x-value. Slope is often represented by the letter m.

Slope Formula

m = (y₂ - y₁) / (x₂ - x₁)

Where (x₁, y₁) and (x₂, y₂) are two points on the line.

The slope is the ratio of the "rise" (vertical change) to the "run" (horizontal change).

Interactive Slope Calculator

Adjust the points below to see how the slope changes. The slope is calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁).

Point 1 (x₁, y₁)

Point 2 (x₂, y₂)

xy(1, 1)(3, 5)
Calculated Slope:
m = (5 - 1) / (3 - 1) = 2.00
For every 1 unit increase in x, y changes by 2.00 units.

Types of Slopes

Positive Slope

When m > 0, the line goes up from left to right.

Negative Slope

When m < 0, the line goes down from left to right.

Zero Slope

When m = 0, the line is horizontal.

Undefined Slope

When the line is vertical, the slope is undefined.

Real-World Applications of Slope

Ramps and Inclines

The slope of a ramp or incline is often expressed as a percentage or ratio. For example, a 10% grade means that for every 100 horizontal units, the elevation changes by 10 units.

Architects and engineers use slope calculations to design wheelchair ramps, roads, and staircases that are safe and meet building codes.

Horizontal DistanceRise

Rate of Change

In many real-world scenarios, slope represents a rate of change:

  • In economics, the slope of a demand curve shows how quantity demanded changes with price.
  • In physics, the slope of a distance-time graph represents velocity.
  • In business, the slope of a cost function shows how costs change with production volume.
  • In medicine, the slope of a patient's temperature chart can indicate the progression of an illness.

Related Topics

Practice Slope Calculations

Test your knowledge with our interactive slope practice exercises.

Start Practice

Slope Formula

m = (y₂ - y₁) / (x₂ - x₁)

Use our calculator to find the slope between any two points.

Open Calculator

Key Concepts

  • 1
    Slope measures the steepness of a line and is calculated as rise over run.
  • 2
    Positive slopes go up from left to right; negative slopes go down from left to right.
  • 3
    Horizontal lines have a slope of zero; vertical lines have an undefined slope.
  • 4
    Parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals of each other.
  • 5
    In real-world applications, slope often represents a rate of change.