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

Thus there is a constant rate of change for $f$, precisely $\frac 1 2$, the value of $m$.