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 : $A = 2:  q(x) = 2x^2$
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 $A>0$ (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 $q$.

Example QF.DID.0 Dynamic 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 function.
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.