# Slope as Rate of Change

The slope of a line represents the rate of change of the dependent variable with respect to the independent variable.

The rate of change between two points on a graph is the ratio of the amount of change in the dependent variable to the amount of change in the independent variable. The values of the dependent variable, typically graphed on the $y$-axis, are determined by (dependent on) the values of the independent variable. The independent variable, typically graphed on the $x$-axis, determines the values of the dependent variable. So, rates of change are given as a ratio of the change in $y$ to the change in $x$ between two points.

A line has a constant rate of change called its slope. The slope $m$ of a line is the ratio of the rise to the run and is the same between any two points on the line:
$m=\frac{\text{Rise}}{\text{Run}}$
The rise is the change in $y$, or the vertical change between two points. The run is the change in $x$, or the horizontal change between the two points. For a line that passes through the points $(x_{1}, y_{1})$ and $(x_{2},y_{2})$, the slope formula is written as:
$m =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
Positive Slope Negative Slope Zero Slope Undefined Slope
As the $x$-values increase, the $y$-values increase. Think of a positive slope as riding a bike up a hill. The rate of change in height is positive. As the $x$-values increase, the $y$-values decrease. Think of a negative slope as riding a bike down a hill. The rate of change in height is negative. As the $x$-values increase, the $y$-values do not change. Think of a zero slope as riding a bike on a flat surface. The rate of change in height is zero. The numerator of the slope formula is zero, while the denominator is nonzero. The line is vertical, so there is no change in $x$-values. No one can ride a bike on a vertical surface, so the slope is undefined. There is only one value of $x$. Thus, there is no rate of change. The denominator of the slope formula is zero, so the value is undefined.