Example LF.2.3: $m = \frac 1 2; b = 1: f (x) = \frac 1 2 x + 1$

Draw a mapping diagram yourself or use the diagram created with GeoGebra to explore the diagram further.
Given a point / number, $x$, on the source line, there is a unique arrow  meeting the target line at the point / number, $\frac 1 2x + 1$, which corresponds to the linear function's value for $x$.

Notice that for each unit increase in $x$ there is a corresponding increase by $\frac 1 2$ units in the  value of $f(x)$.
Thus there is a constant rate of change for $f$, precisely $\frac 1 2$,  the value of $m$.