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Interpreting Quadratic Functions in Factored Form

Interpreting Quadratic Functions in Factored Form 1.1 - Solution

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Recall the factored form of a quadratic function. y=a(xp)(xq)\begin{gathered} y=a(x-p)(x-q) \end{gathered} In this form, where aa \neq 0,0, the x-x\text{-}intercepts are pp and q.q.

Let's consider the factored form of our function. y=-2(x2)(x5)\begin{gathered} y={\color{#FF0000}{\text{-}2}}(x-{\color{#0000FF}{2}})(x-{\color{#009600}{5}}) \end{gathered} We can see that a=-2,{\color{#FF0000}{a}}={\color{#FF0000}{\text{-}2}}, p=2,{\color{#0000FF}{p}}={\color{#0000FF}{2}}, and q=5.{\color{#009600}{q}}={\color{#009600}{5}}. Therefore, the x-x\text{-}intercepts are 2{\color{#0000FF}{2}} and 5.{\color{#009600}{5}}. Next, we will find the axis of symmetry, which is the vertical line halfway between (p,0)(p,0) and (q,0).(q,0). Since we know that p=2p=2 and q=5,q=5, the axis of symmetry of our parabola is halfway between (2,0)(2,0) and (5,0).(5,0). x=p+q2x=2+52=72=3.5\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.x=3.5.