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
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.)
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 $f$.
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
The value of $b$ does not effect whether $f$ is increasing or
Note: The position of the focus point is consistent
with the effect of $m$ on whether the function is increasing or