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$, the following fundamental observations should make sense in the mapping diagrams: • Extreme Value of$f$: The extreme value for$f$will occur at$x=h$. This value will be$f(h) = A(h-h)^2 +k = k$. • When$A \gt 0$the value at$x=h$, that is,$k$will be the smallest (minimum) value for$f$. • When$A < 0$the value at$x=h$, that is,$k$will be the largest (maximum) value for$f$. • Magnification, The effect of$A$: The value of$A$can be seen by looking at the change in the value of$f$from$x=h$to$x= h+1$.$f(h) = k$while$f(h+1) = A +k$. Though this change is not a constant for all changes in$x$, it indicates the magnification that will effect all other changes for$f$from the core function of$x^2$. It will be greater for greater values of$|A|$, and with determine whether it is an increase of decrease by whether$A \gt 0$or$A < 0$. On the graph this is usually described as controlling the bowl shape of the parabolic graph of$f$. • Symmetry of$f$with respect to$x=h$: The values$f$at$x=h+a$and$x=h-a$will be equal since$((h+a)-h)^2 + k= a^2+k =(-a)^2 +k =((h-a)-h)^2$+k. Thus$f$takes on equal values for numbers symmetrically located by either adding/subtracting the same number to/from$h$in the domain. You can use the next dynamic example to see the effects of the quadratic coefficient and adding constants before and after$x^2\$ simultaneously on a mapping diagram and a graph.

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