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

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 $q$.

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

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 $q$.