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Interpreting Quadratic Functions in Factored Form

Interpreting Quadratic Functions in Factored Form 1.21 - Solution

We want to write the equation of the parabola that passes through the points $(\text{-} 5,0),$ $(\text{-} 1,0),$ and $(\text{-}4,3).$ Note that the $y\text{-}$coordinate of the first two points is $0,$ and therefore the $x\text{-}$intercepts of the graph are $\textcolor{#ff8c00}{\text{-} 5}$ and ${\color{#FF0000}{\text{-}1}}.$ Let's recall the factored form of a quadratic function. $\begin{gathered} y=a(x-p)(x-q) \end{gathered}$ In this form, $p$ and $q$ are the intercepts. Therefore, we can already partially write our equation. $\begin{gathered} y=a(x-(\textcolor{#ff8c00}{\text{-} 5}))(x-({\color{#FF0000}{\text{-}1}})) \\ \Updownarrow \\ y=a(x+5)(x+1) \end{gathered}$ Since the parabola passes through the point $(\text{-}4,3),$ we can substitute $\text{-}4$ for $x$ and $3$ for $y$ in the above equation, and solve for $a.$
$y=a(x+5)(x+1)$
${\color{#009600}{3}}=a({\color{#0000FF}{\text{-}4}}+5)({\color{#0000FF}{\text{-}4}}+1)$
Solve for $a$
$3=a(1)(\text{-}3)$
$3=a(\text{-}3)$
$\text{-}1=a$
$a=\text{-} 1$
Knowing that $a=\text{-} 1,$ we can write the full equation of the parabola. $\begin{gathered} y=\text{-}1(x+5)(x+1) \quad \Leftrightarrow \quad y=\text{-} (x+5)(x+1) \end{gathered}$