Secant Lines

Definition

A secant line is a straight line joining two points on a function or curve. It provides a linear approximation of the curve between those two points.

Why It Matters

Secant lines serve as the geometric foundation for understanding rates of change. By shrinking the interval between the two points, the secant line approaches the tangent line, bridging the gap between algebra (slope) and calculus (derivative).

Core Concepts

• Slope of the Secant Line: The slope of the secant line passing through the points $P(x_1, f(x_1))$ and $Q(x_2, f(x_2))$ represents the average rate of change of the function over that interval.
• Formula for the Slope: $m_{sec} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$
  • How to read: “The slope of the secant line is equal to the difference f of x two minus f of x one, all over the difference x two minus x one.”
  • Meaning: It calculates the average rate of change between points P and Q.

Connected Concepts

• Average Rate of Change
• Derivative as a Rate of Change
• Slope
• Lines