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 $[m,b]$.

Definition FP  Focus Point

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.

Example LF.3.1 : $m =-2; b = 1: f(x) = -2x + 1$
Example LF.3.2 : $m = 2; b = 1: f(x) = 2x + 1$
Example LF.3.3 : $m =\frac 1 2 ; b = 1: f(x) = \frac 1 2x + 1$
Example LF.3.4 : $m = 0; b = 1: f(x) = 0 x + 1$
Example LF.3.5 : $m = 1; b = 1: f(x) = x + 1$

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.