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.

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

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 $f$.

The value of $b$ does not effect whether $f$ is increasing or decreasing.