Tangents, Derivatives and Differentiation

The Slope of a Line

Let P = (x₁,y₁) and Q =(x₂,y₂) be two points on the Cartesian plane. The slope of the straight line passing through P and Q is

This is sometimes written as

where and and we say "the slope is equal to the change in y relative to the change in x" .

Note that m = tan θ, where θ is the angle of inclination.

Note also that if m > 0, the line is rising (or increasing). If m < 0, the line in falling (or decreasing) and if m = 0 the line is horizontal.

The slope of the line determined by two points is used in many practical situations, such as finding the average velocity (or change in distance over time) and finding the average marginal cost (or change in cost per unit).

A vertical line has undefined slope because all points on the line have the same x-coordinate . As a result the formula used for slope has a denominator of 0, which makes the slope undefined.

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