Rise Over Run
AlgebraRise over run is a way to describe slope: the vertical change (rise) divided by the horizontal change (run) between two points on a line.
Formula
\text{slope} = \frac{\text{rise}}{\text{run}}
Definition
Rise over run is a simple way to find slope. "Rise" is how many units you move up or down, and "run" is how many units you move right. Slope $=$ rise $/$ run.
Example
If you move up $4$ units and right $2$ units between two points, the slope is $4/2 = 2$. If you move down $3$ and right $1$, the slope is $-3/1 = -3$.
Key Insight
Imagine walking along a hill: rise is how high you climb, run is how far forward you walk. A steep hill has a big rise with a small run.
Definition
Rise over run expresses slope as the ratio of the vertical change ($\Delta y$) to the horizontal change ($\Delta x$) between any two points on a line. Rise is positive going up and negative going down. Run is always measured to the right (positive direction).
Example
On a graph, move from $(2, 1)$ to $(5, 7)$: rise $= 7 - 1 = 6$, run $= 5 - 2 = 3$, slope $= 6/3 = 2$. This matches the slope formula $(y_2-y_1)/(x_2-x_1)$.
Key Insight
Rise over run makes slope visual. You can read the slope directly from a graph by counting squares: pick two clear lattice points on the line and count up (or down) and over.
Definition
Rise over run is the geometric interpretation of the difference quotient $(f(x + h) - f(x))/h$ in the limit as $h$ approaches $0$, which defines the derivative. For a linear function $f(x) = mx + b$, the difference quotient equals $m$ for all $h$, confirming that slope is constant everywhere on a line.
Example
$f(x) = 3x + 1$: $(f(x+h) - f(x))/h = (3(x+h) + 1 - 3x - 1)/h = 3h/h = 3$. The rise over run is exactly $3$, independent of $x$ and $h$.
Key Insight
The difference quotient formula for derivative is rise over run in the limit. Linear functions are the only ones where this ratio is constant, which is why they have a single well-defined slope everywhere.