Lesson 4-2: Slope of a Line

Property

The slope of a line is a rate of change that measures the steepness of the line.

It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate. In symbols:

Examples

• A line passes through the points $(3, 5)$ and $(7, 13)$. Its slope is $m = \frac{13-5}{7-3} = \frac{8}{4} = 2$.
• For a line containing points $(-2, 9)$ and $(4, 6)$, the slope is $m = \frac{6-9}{4-(-2)} = \frac{-3}{6} = -\frac{1}{2}$.
• The slope of a line passing through $(100, 50)$ and $(120, 40)$ is $m = \frac{40-50}{120-100} = \frac{-10}{20} = -0.5$.

Property

Positive slopes correspond to lines that increase from left to right.
Negative slopes correspond to lines that decrease from left to right.
The larger the absolute value of the slope, the steeper the graph.

Examples

• A line with slope $m = 2$ is steeper than a line with slope $m = \frac{1}{3}$ because $|2| > |\frac{1}{3}|$.

• A line with slope $m = -3$ is steeper than a line with slope $m = -1$ because $|-3| > |-1|$. Both lines slant downwards.