Linear Functions and Their Graphs

A linear function is a function that is changing at a constant rate. That is, the change $\Delta y$ of the output is proportional to the change $\Delta x$ of the input. When we graph a linear function, we get a straight line, and the proportionality constant $m$ is the slope of the line. That is,

$m$ = slope = rate of change = conversion factor between $\Delta x$ and $\Delta y$.

There are several ways to write the formula for a linear function, or equivalently the equation of a straight line:

$m = \Delta y / \Delta x$ = rise/run. (Definition of slope).
$(y-y_0) = m (x-x_0)$. (Point-slope formula)
$y = mx + b$. (Slope-intercept form)
$\Delta y = m \Delta x$. (Microscope equation)
$\Delta x = \Delta y/m$.

You should understand how to convert from one form to another, and what the different parameters mean.