Slope Calculator

Slope is a measure of the steepness and direction of a line, representing the rate of change between two variables. Core Definition: Slope (m) = Rise / Run = Vertical Change / Horizontal Change Standard Formula: For two points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁) / (x₂ - x₁) Where: • m = slope (gradient) • (x₁, y₁) = coordinates of first point • (x₂, y₂) = coordinates of second point • y₂ - y₁ = rise (vertical change) • x₂ - x₁ = run (horizontal change) Types of Slope: 1. Positive Slope (m > 0): • Line rises from left to right • As x increases, y increases • Example: m = 2, line goes upward 2. Negative Slope (m < 0): • Line falls from left to right • As x increases, y decreases • Example: m = -3, line goes downward 3. Zero Slope (m = 0): • Horizontal line • y-value constant, no vertical change • Equation form: y = b 4. Undefined Slope: • Vertical line • x-value constant, division by zero • Equation form: x = a • Cannot be expressed as a number Slope-Intercept Form: y = mx + b Where: • m = slope • b = y-intercept (where line crosses y-axis) Point-Slope Form: y - y₁ = m(x - x₁) Useful when you know: • One point (x₁, y₁) • The slope m Key Properties: 1. Parallel lines have equal slopes: m₁ = m₂ 2. Perpendicular lines have negative reciprocal slopes: m₁ × m₂ = -1 3. Steeper lines have larger absolute values of slope 4. Slope is constant for straight lines Interpretation Examples: • m = 2: For every 1 unit right, go up 2 units • m = -1/2: For every 2 units right, go down 1 unit • m = 0: No rise, completely flat Applications: Road grade calculations, roof pitch, economics (marginal cost/revenue), physics (velocity, acceleration), rate of change analysis, trend lines in data, and graphing linear relationships.