3-2 Rate of Change and Slope

Property

The slope of a line gives us the rate of change of one variable with respect to another.

Formula for slope:

Examples

Find the slope between $(-1, 4)$ and $(3, -2)$: $m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}$.

Property

The slope of the line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$. This is the slope formula. The slope is the difference in the y-coordinates divided by the difference in the x-coordinates.

Examples

• For the points $(2, 5)$ and $(4, 11)$, the slope is $m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3$.
• For the points $(-3, 6)$ and $(1, 4)$, the slope is $m = \frac{4 - 6}{1 - (-3)} = \frac{-2}{4} = -\frac{1}{2}$.
• For the points $(5, -1)$ and $(-2, 3)$, the slope is $m = \frac{3 - (-1)}{-2 - 5} = \frac{4}{-7} = -\frac{4}{7}$.

Explanation

The slope formula is a tool to find a line's steepness without a graph. It calculates the rise by subtracting y-values ($y_2 - y_1$) and the run by subtracting x-values ($x_2 - x_1$), then divides them.