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.
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 ($x = h$).