Interpreting Quadratic Functions in Vertex Form

Let's start by recalling the vertex form of a quadratic function.

Since we want our function to have a vertex at $({\color{#0000FF}{1}},{\color{#009600}{3}})$ we can wirte it in vertex form using ${\color{#0000FF}{h}}={\color{#0000FF}{1}}$ and ${\color{#009600}{k}} = {\color{#009600}{3}}.$ Note that $a$ can take any value as it does not affect the location of the vertex. For simplicity, we will use $\textcolor{#FF00FF}{a} = \textcolor{#FF00FF}{1}.$ $\begin{gathered} f(x) = \textcolor{#FF00FF}{1}(x-\textcolor{#0000FF}{1})^2 +\textcolor{#009600}{3} \quad \Leftrightarrow \quad f(x)=(x-1)^2+3 \end{gathered}$ Notice that we could have chosen any nonzero value for $a.$ Therefore, there are infinitely many quadratic functions whose vertex is located at $(1,3).$ This is just one example.