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 have seen the importance these two forms in the work in Subsection QF.ID .

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.

Two simple examples.
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.
Example QF.3.1 : \$ f(x) = x^2; g(x) = x-2\$
Example QF.2.2 :\$ f(x) = 2x ; g(x) = x^2\$
Example QF.COMP.2 :\$ f(x) = x - 2; g(x) = x^2\$

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}\$.

Symmetry of Quadratic Functions: Suppose \$f(x) =A (x-h)^2 + k\$.

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\$.

Example QF.DSYMM.0 Dynamic Visualization of Symmetry for Quadratic Functions: Graphs, and Mapping Diagrams