Consider the quadratic function $q(x) = Ax^2 + Bx +C$. ($A \ne 0$)

This form for a quadratic function is usually described as the "standard polynomial" form of $q$.

The number $A$ is referred to as the "leading coefficient" and the value of $q(0) = C$ gives the second coordinate of the Y-intercept, $(0,C)$ for the graph of $q$.
The Vertex Form: There is one other very useful form for the quadratic function $q$ that uses $A$, the leading coefficient of $q$, and two number $h$ and $k$ to evaluate $q(x)$.
In this form, $q$ is evaluated by $q(x) = A(x-h)^2 +k $.

This form is called the "vertex" form of $q$. As we will see in QF.MA, the value $q(h) = k$  is an extreme value for $q$.

Connection between the Standard and Vertex Forms.
Expanding the vertex form expression gives $q(x) =A(x^2 - 2hx +h^2) + k = Ax^2  - 2Ahx + (Ah^2 +k)$.
From this expression we can see the connection between the coefficients in the standard form and the numbers $h$ and $k$ of the vertex form.

In the standard form the coefficient of $x$ is $B$, but from the vertex form it is  $-2Ah$.
We can set these equal: $B = -2Ah$ and since $A \ne 0$, we have $h = -\frac B {2A}$.

Since in the vertex form it is clear that $q(h) = k$, we have $ k = A h^2 + Bh + C$ while $C = Ah^2 +k$.
Thus - If we know $A,B$, and $C$ for the standard form, we can determine $h$ and $k$ of the vertex form, and conversely, if we know $A, h$ and $k$ of the vertex form we can find $B$ and $C$ for the standard form. We summarize this connection in the following
Theorem QF.COMP
The two forms, as well as how to find the values of $A,B$ and $C$ from three values for $q$, are covered thoroughly in most algebra texts.

In later subsections we explore how mapping diagrams for quadratic functions help us visualize these forms and understand further the meaning of the shape of the graph of $q$.
Example QF.FORM.1 : Illustrating the leading coefficient = $A=2$ and the Y-intercept is $-4$.
Example QF.FORM.2 : Illustrating leading coefficient $A=2$ and extreme value when $q(-1)= 3$.
Example QF.FORM.3 : Illustrating a quadratic function $q$ with extreme value $q(1)= 2$  and  $q(2) = 1$.
Example QF.FORM.4 : Illustrates a quadratic function as the composition of the core functions  $f_{+ 3}\circ f_{*2}\circ q\circ f_{ -1}$.
Example QF.DVFORM.0 : Dynamic Visualization of vertex form for a quadratic function: Graphs and Mapping Diagrams

Example QF.DSFORM.0 : Dynamic Visualization of standard form for a quadratic function: Graphs and Mapping Diagrams
To be added ?... A dynamic example to visualize finding $A$ and the vertex form of a quadratic function with a mapping diagram of $f$  and the parabola in the graph of $f$.