Interpreting Quadratic Functions in Vertex Form

We want to identify the vertex and the axis of symmetry of the graph of given quadratic function. To do so, we will first express it in vertex form where $a,$ $h,$ and $k$ are either positive or negative numbers. $\begin{gathered} y=\text{-}(x-4)^2-3 \quad \Leftrightarrow \quad y=\text{-}1\cdot\big(x-4\big)^2+(\text{-}3) \end{gathered}$ It is important to note that we do not need to graph the parabola to identify the desired information. Let's compare the general formula for the vertex form with our equation. $\begin{aligned} \textbf{General formula: }y=&\ \phantom{\text{-}}{\color{#FF0000}{a}}(x-{\color{#0000FF}{h}})^2 +\phantom{\text{-}\text{-}}\textcolor{magenta}{k} \\ \textbf{Equation: }y=&\ {\color{#FF0000}{\text{-}1}}\big(x-{\color{#0000FF}{4}}\big)^2+(\textcolor{magenta}{\text{-}3}) \end{aligned}$ We can see that ${\color{#FF0000}{a}}={\color{#FF0000}{\text{-}1}},$ ${\color{#0000FF}{h}}={\color{#0000FF}{4}},$ and $\textcolor{magenta}{k}=\textcolor{magenta}{\text{-} 3}.$

Vertex

The vertex of a quadratic function written in vertex form is the point $({\color{#0000FF}{h}},\textcolor{magenta}{k}).$ For this exercise, we have ${\color{#0000FF}{h}}={\color{#0000FF}{4}}$ and $\textcolor{magenta}{k}=\textcolor{magenta}{\text{-}3 }.$ Therefore, the vertex of the given equation is $({\color{#0000FF}{4}},\textcolor{magenta}{\text{-} 3}).$

Axis of Symmetry

The axis of symmetry of a quadratic function written in vertex form is the vertical line with equation $x={\color{#0000FF}{h}}.$ As we have already noticed, for our function, this is ${\color{#0000FF}{h}}={\color{#0000FF}{4}}.$ Thus, the axis of symmetry is the line $x={\color{#0000FF}{4}}.$