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Interpreting Quadratic Functions in Factored Form

Interpreting Quadratic Functions in Factored Form 1.21 - Solution

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We want to write the equation of the parabola that passes through the points (-5,0),(\text{-} 5,0), (-1,0),(\text{-} 1,0), and (-4,3).(\text{-}4,3). Note that the y-y\text{-}coordinate of the first two points is 0,0, and therefore the x-x\text{-}intercepts of the graph are -5\textcolor{#ff8c00}{\text{-} 5} and -1.{\color{#FF0000}{\text{-}1}}. Let's 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, pp and qq are the intercepts. Therefore, we can already partially write our equation. y=a(x(-5))(x(-1))y=a(x+5)(x+1)\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 (-4,3),(\text{-}4,3), we can substitute -4\text{-}4 for xx and 33 for yy in the above equation, and solve for a.a.
y=a(x+5)(x+1)y=a(x+5)(x+1)
3=a(-4+5)(-4+1){\color{#009600}{3}}=a({\color{#0000FF}{\text{-}4}}+5)({\color{#0000FF}{\text{-}4}}+1)
Solve for aa
3=a(1)(-3)3=a(1)(\text{-}3)
3=a(-3)3=a(\text{-}3)
-1=a\text{-}1=a
a=-1a=\text{-} 1
Knowing that a=-1,a=\text{-} 1, we can write the full equation of the parabola. y=-1(x+5)(x+1)y=-(x+5)(x+1)\begin{gathered} y=\text{-}1(x+5)(x+1) \quad \Leftrightarrow \quad y=\text{-} (x+5)(x+1) \end{gathered}