**Symmetry and Composition of Quadratic Functions**

Consider the quadratic
function $f(x) = Ax^2 + Bx + C$ (Standard Form) $= A (x-h)^2
+ k$ (Vertex Form).

We continue her to explore how the Vertex Form in particular can
be used to understand the symmetry of the
quadratic as a composition
of core functions.

In these examples (two from QF.MA) we use GeoGebra to visualize the composition for a single linear core function with the quadratic core function.

Compositions are a key to understanding the symmetry of quadratic functions. The following key result on quadratic functions as compositions is very useful.

This result says **quadratic functions are built by composing
three core linear functions with the core squaring function:**
$q(x)= x^2$ with linear functions, $f_{-h}, f_k$, and $f_{*A}$.

Here is an example from QF.FORM that visualizes the theorem.

** Example** QF.FORM.4 Suppose $f$
is a quadratic function with leading coefficient $ =A=2$ and
extreme value $f(1)=3$. Find the vertex form of the quadratic
function. Visualize $f$ with a mapping diagram that illustrates
the function as the composition of the four core functions
$f_{+3}∘f_{∗2}∘q∘f_{−1}$.

Understanding $f$ with compositions and mapping diagrams explains why quadratic functions always have even symmetry with respect to an axis ($x = h$).

You can use this next dynamic example with GeoGebra to investigate further the symmetry of a quadratic in a mapping diagram of $f$.