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

- Square $x$ to form $x^2$.
- Multiply the result of step 1 by the magnification factor $A$ to form $Ax^2$.
- Perform the linear function steps to form $Bx+C$.
- 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, $A$

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

**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****$A < 0$****$k$****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.