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.

Definition ID Increasing/Decreasing
Definition EV 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\$. ["bowl up"]
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 functions.)

Example QF.2.1 : \$A =-2: q(x) = -2x^2\$
Example QF.2.2 : \$A = 2:  q(x) = 2x^2\$