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Recall the factored form of a quadratic function. $\begin{gathered} y=a(x-p)(x-q) \end{gathered}$ In this form, where $a$ $\neq$ $0,$ the $x\text{-}$intercepts are $p$ and $q.$
Let's consider the factored form of our function. $\begin{gathered} y={\color{#FF0000}{\text{-}2}}(x-{\color{#0000FF}{2}})(x-{\color{#009600}{5}}) \end{gathered}$ We can see that ${\color{#FF0000}{a}}={\color{#FF0000}{\text{-}2}},$ ${\color{#0000FF}{p}}={\color{#0000FF}{2}},$ and ${\color{#009600}{q}}={\color{#009600}{5}}.$ Therefore, the $x\text{-}$intercepts are ${\color{#0000FF}{2}}$ and ${\color{#009600}{5}}.$ Next, we will find the axis of symmetry, which is the vertical line halfway between $(p,0)$ and $(q,0).$ Since we know that $p=2$ and $q=5,$ the axis of symmetry of our parabola is halfway between $(2,0)$ and $(5,0).$ $\begin{gathered} x=\dfrac{p+q}{2}\quad \Rightarrow \quad x=\dfrac{{\color{#0000FF}{2}}+{\color{#009600}{5}}}{2}=\dfrac{7}{2}=3.5 \end{gathered}$ We found that the axis of symmetry is the vertical line $x=3.5.$