Consider the linear function \$f(x) = mx + b\$.

When \$m = 0\$, the function  has the same constant value \$b\$ for any \$x\$. "The function \$f\$ is a constant function."
When \$m \ne 0\$ , the values of the  linear function \$f\$ vary in predictable ways depending on whether \$m>0\$ or \$m<0\$.
This is apparent by reviewing the mapping diagrams along with the graphs in some of our previous examples.
But first we define the key concepts: increasing and decreasing.

Definition ID Increasing/Decreasing

Review of two previous examples.
Notice how the graph and mapping diagram visualize the fact that for linear functions, if \$m >0\$ then \$f\$ is an increasing function while if \$m<0\$ then \$f\$ is a decreasing function ( and the converses are also true for linear functions.)

Example LF.4.1 : \$m =-2; b = 1: f(x) = -2x + 1\$
Example LF.4.2 : \$m = 2; b = 1:  f(x) = 2x + 1\$

You can use this next dynamic example to investigate visually the effects of the linear coefficient and the constant term simultaneously on whether the function is increasing or decreasing in a mapping diagram  of \$f\$ and the line in the graph of \$f\$.

Example LF.DID.0 Dynamic Visualization of Increasing and Decreasing for Linear Functions: Graphs, and Mapping Diagrams

Reminder: If \$m >0\$ then \$f\$ is an increasing function while if \$m<0\$ then \$f\$ is a decreasing function.
The value of \$b\$ does not effect whether \$f\$ is increasing or decreasing.