Consider the quadratic function $q(x) = Ax^2 +Bx+ C$.

When $A= 0$, the function is not a quadratic function, but is a linear function: $q(x) = Bx + C$.

When $A \ne 0$ , the values of the quadratic function $q$ vary in predictable ways depending on whether $A>0$ or $A<0$.This is apparent by reviewing the mapping diagrams along with the graphs in some of our previous examples.

But first we review the key concepts: increasing and decreasing- and an extreme value for a function.

Notice how the graph and mapping diagram visualize the following fact that for quadratic functions that is justified in all texts that cover quadratic functions.

(

Because of the previous fact, every quadratic function will have an extreme value. On the graph of the quadratic function, this is visualized by the "vertex" of the parabolic curve. On the mapping diagram this is visualized by the fact that the arrows all land above (or below) the value $f(c)$ on the target axis when $A>0$ (when $A<0$).

You can use this next dynamic example to investigate visually the effects of the quadratic coefficient on the increasing and decreasing shape of quadratic a mapping diagram of $q$ and the parabolic curve in the graph of $q$.

The values of $B$ and $C$ do not effect the shape of $q$.