A quadratic function is usually presented in a standard form $f(x) = Ax^2 + Bx + C$ instead of $f(x) = A(x-h)^2 + k$. The algebra that goes with identifying these two forms is studied extensively  in most texts.

By simply expanding $f(x) = A(x-h)^2 + k$ we find $f(x) = Ax^2-2Axh  + h^2 + k$. So for these two forms to be giving the same function $f$, $B = -2Ah$. So $h = -\frac B {2A}$. Since the extreme value for $f$ will occur at $x=h$, he extreme value for $f$ will occur at $x= -\frac B A$. This is also the center for the symmetry of $f$ in its domain for the mapping diagram and the axis of symmetry for the graph of $f$.
The relation of $k$ to the standard form coefficients is found simply by evaluation of $f$ at $x=-\frac B {2A}$ so
$ k = f(-\frac B {2A}) = A(-\frac B {2A})^2  + B*-\frac B {2A} + C = \frac {B^2} {4A} - \frac{B^2} {2A} + C = \frac {4AC-B^2} {4A}.

In considering the quadratic functions in the standard form, we can read the instructions for the function as a commands:
  1. Square $x$ to form $x^2$.
  2. Multiply the result of step 1 by the magnification factor $A$ to form $Ax^2$.
  3. Perform the linear function steps to form $Bx+C$.
  4. Add the result of step 2 to the result of step 3 to form the quadratic function $f(x) = Ax^2 +Bx +C$.

Note that for Example QF.0, $x^2-2x-3 = (x-1)^2 -4$ so $A=1$, $h=1$ and $k = -4$.
You should  review Example QF.0  to explore for yourself how the numbers $A$, $h$,  and $k$ are realized in the features of the mapping diagram.

Before further discussion we'll examine some simple and important examples.

Example QF.1.1 : $q(x) = x^2$.  " Square x" : $x^2$  The core quadratic function.
Example QF.1.2 : $q(x) = x^2+ 2$.  "Added value after $x^2$ ": $2$
Example QF.1.3 : $q(x) = (x-2)^2$.  "Added value before $(...)^2$ ": $x - 2$ 
Example QF.1.4 : $ q(x) = 2x^2$.    "Constant multiple": $2$ 

Now that you've looked a some simple examples here are four more [important] examples for the quadratic function $q(x) = Ax^2 $.
These examples illustrate the effect of  the quadratic coefficient, $A$

Example QF.2.1 : $A =-2; q(x) = -2x^2$
Example QF.2.2 : $A = 2;  q(x) = 2x^2 $
Example QF.2.3 : $A =\frac 1 2 ;  q(x) = \frac 1 2x^2$
Example QF.2.4 : $A = -\frac 1 2;  q(x) = -\frac 1 2x^2$

You can consider the effect of other adding "values after $x^2$ "in the exercises or with the dynamic Example QF.DV.0.
"Adding value before $(...)^2$ " is worth a few more examples treated as compositions of $g(x) =x - h$ before $f(x) = x^2$
Example QF.3.1 : $h =2; q(x) = (x-2)^2$
Example QF.3.2 : $h = -2;  q(x) = (x+2)^2 $

From this construction and understanding of the core quadratic function  $q(x) = x^2$, the following fundamental observations should make sense in the mapping diagrams:

You can use the next dynamic example to see the effects of the quadratic coefficient and adding constants before and after $x^2$ simultaneously on a mapping diagram and a graph.

Example QF.DV.0 Dynamic Visualization for Quadratic Functions: Graphs and Mapping Diagrams