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.

**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:

$m=\frac{\text{Rise}}{\text{Run}}$

**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. |