Consider the linear function $f(x) = mx + b$. This is studied
extensively with its graph in most texts.
The linear function $f$ determines a non-vertical line with
its graph.
Similarly, as long as $m \ne 1$ , the linear function $f$ determines
a specific point on the mapping diagram which we will call the
focus point of $f$. Since this point is determined by
the function parameters $m$ and $b$, we will designate this point on
the mapping diagram with $[m,b]$.
Comment: It is an exercise in
geometry to prove that when $m \ne 1$, the focus point of $f$,
is unique and thus the focus point of $f$ is well defined.
Review of some of the previous examples.
You will notice that the arrows in these examples where $m \ne 1$
meet at a common point- the focus point for that linear function,
and when $m = 1$, the arrows and lines are all parallel.
You can use this next dynamic example to
investigate the effects of the linear coefficient and the constant
term simultaneously on the focus point in a mapping diagram of
$f$ and the line in the graph of $f$.
Example LF.DFP.0 Dynamic Visualization of Focus Point for
Linear Functions: Graphs, and Mapping Diagrams
Note: The value of $m$ places the focus
point by its distance from the two axes.
For a fixed vale of $m$ the value of $b$ places the focus on a line
determined by $m$ parallel to the domain (source) axis.