Consider the linear function $f(x) = mx + b$. This is studied extensively with its graph in most texts.
We  will examine mapping figures for an alternative visualization of  the key features of $f $.-

You should first review Example LF.0  to explore for your self how the numbers $m$ and $b$ are realized in the features of the mapping diagram.

Before further discussion of  we'll examine some simple and important examples.

Example LF.1.1 : $f(x) = 3$.  "Constant value": $3$
Example LF.1.2 : $f(x) = x + 2$.  ""Added value": $2$
Example LF.1.3 : $ f(x) = 2x$.    "Scalar Multiple": $2$ 


Now that you've looked a some simple examples here are five more [important] examples for the linear function $f(x) = mx + b$. Each has the same "constant", $1$, so these examples illustrate the effect of  the linear coefficient, $m$

Example LF.2.1 : $m =-2; b = 1: f(x) = -2x + 1$
Example LF.2.2 : $m = 2; b = 1:  f(x) = 2x + 1$
Example LF.2.3 : $m =\frac 1 2 ; b = 1: f(x) = \frac 1 2x + 1$
Example LF.2.4 : $m = 0; b = 1: f(x) = 0 x + 1$
Example LF.2.5 : $m = 1; b = 1:  f(x) = x + 1$


Here is a link to a spreadsheet for exploring the effects of $m$ and $b$ on the mapping diagram for a linear function.

You can use the following dynamic example to see the effects of the linear coefficient simultaneously on a mapping diagram and a graph.
Example LF.DA.0 Dynamic Visualization for Linear Functions: Graphs, and Mapping Diagrams
 
Still to investigate:??
Parallel Lines: m=m'
Perpendicular Lines
                m*m' = -1

See also previous work by Martin Flashman, Yoon Kim, and Ken Yanosko, introducing mapping diagrams with dynamics: Visualizing Functions Pages 2-6