# In a graph, what does rise over run equal?

Jun 24, 2015

In a word 'slope'. To be more specific, 'average slope'.

#### Explanation:

If a straight line passes through two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ where ${x}_{2} > {x}_{1}$. then the run is $\left({x}_{2} - {x}_{1}\right)$, the rise is $\left({y}_{2} - {y}_{1}\right)$ and the slope $m$ of the line is defined as:

$m = \frac{\Delta y}{\Delta x} =$ run / rise $= \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

If instead of a straight line, we have a function $f \left(x\right)$ which is continuous and otherwise well-behaved over the interval $\left[{x}_{1} , {x}_{2}\right]$ and $f \left({x}_{1}\right) = {y}_{1}$ and $f \left({x}_{2}\right) = {y}_{2}$ then the average slope of $f \left(x\right)$ over the interval $\left[{x}_{1} , {x}_{2}\right]$ is also $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

If you are familiar with the terminology, we are basically evaluating the integral of the derivative of $f \left(x\right)$ over the interval $\left[{x}_{1} , {x}_{2}\right]$, then dividing by the length of the interval. This is like adding up the slopes at each point and dividing by the number of measurements to get the average.