Slope

Definition

Slope ($m$) is a numerical value that describes both the direction and the steepness of a line or curve.

Why It Matters

Slope is the geometric basis for the derivative in calculus. It represents rate of change—how much one variable changes relative to another.

Core Concepts

• Formula: For a line passing through $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • How to read: “m equals y two minus y one over x two minus x one.”
  • Meaning / when to use: Rise over run—how much $y$ changes per unit change in $x$.
• Interpretation:
  • $m > 0 \to$ Rising.
  • $m < 0 \to$ Falling.
  • $m = 0 \to$ Horizontal.
  • Undefined $\to$ Vertical.

Connected Concepts

• Lines
• Cartesian Coordinate Plane
• Distance Formula
• Midpoint Formula
• Analytic Proof Strategies
• Linear Functions