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.
Review of two previous examples.
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.
Fact (The shape of quadratic functions):
Case P(ositive and uP): If $A >0$ then there
is a number $c$ where $f$ is an decreasing function for
all $x\le c$ while $f$ is an increasing function for all $x\ge c$.
Case N(egative and dowN) If $A <0$ then there
is a number $c$ where $f$ is an increasing function for
all $x \le c$ while $f$ is an decreasing function for all $x \ge
c$. ["bowl down"]
(The converses of each of these are also true for quadratic
The Quadratic Function Extreme.
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 $A>0$
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 $q$.
Visualization of Increasing and Decreasing for Quadratic
Functions: Graphs, and Mapping Diagrams
Reminder: If $A >0$ then $q$ is
a bowl up function while if $A<0$ then $q$ is a bowl down
The values of $B$ and $C$ do not effect the shape of $q$.
Note: The position of the extreme point is consistent with
the effect of $A$ on whether the function is bowl up or bowl down.