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Interpreting Quadratic Functions in Factored Form

Interpreting Quadratic Functions in Factored Form 1.11 - Solution

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To draw the graph of the given function, we will follow four steps.

  1. Identify and plot the x-x\text{-}intercepts.
  2. Find and graph the axis of symmetry.
  3. Find and plot the vertex.
  4. Draw the parabola through the vertex and the points where the x-x\text{-}intercepts occur.

Let's go through these steps one at a time.

Identify and Plot the x-x\text{-}intercepts

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. f(x)=-3(x+1)(x+7)f(x)=-3(x(-1))(x(-7))\begin{gathered} f(x)=\text{-}3(x+1)(x+7) \\ \Updownarrow \\ f(x)={\color{#FF0000}{\text{-}3}}(x-({\color{#0000FF}{\text{-} 1}}))(x-({\color{#009600}{\text{-}7}})) \end{gathered} We can see that a=-3,{\color{#FF0000}{a}}={\color{#FF0000}{\text{-}3}}, p=-1,{\color{#0000FF}{p}}={\color{#0000FF}{\text{-}1}}, and q=-7.{\color{#009600}{q}}={\color{#009600}{\text{-}7}}. Therefore, the x-x\text{-}intercepts occur at (-1,0)({\color{#0000FF}{\text{-}1}},0) and (-7,0).({\color{#009600}{\text{-}7}},0).

Find and Graph the Axis of Symmetry

The axis of symmetry is halfway between (p,0)(p,0) and (q,0).(q,0). Since we know that p=-1p=\text{-}1 and q=-7,q=\text{-}7, the axis of symmetry of our parabola is halfway between (-1,0)(\text{-}1,0) and (-7,0).(\text{-}7,0). x=p+q2x=-1+(-7)2=-82=-4\begin{gathered} x=\dfrac{p+q}{2}\quad \Rightarrow \quad x=\dfrac{{\color{#0000FF}{\text{-}1}}+({\color{#009600}{\text{-}7}})}{2}=\dfrac{\text{-}8}{2}=\text{-}4 \end{gathered} We found that the axis of symmetry is the vertical line x=-4.x=\text{-}4.

Find and Plot the Vertex

Since the vertex lies on the axis of symmetry, its x-x\text{-}coordinate is -4.\text{-}4. To find the y-y\text{-}coordinate, we will substitute -4\text{-}4 for xx in the given equation.
f(x)=-3(x+1)(x+7)f(x)=\text{-}3(x+1)(x+7)
f(-4)=-3(-4+1)(-4+7)f({\color{#0000FF}{\text{-}4}})=\text{-}3({\color{#0000FF}{\text{-}4}}+1)({\color{#0000FF}{\text{-}4}}+7)
f(-4)=-3(-3)(3)f(\text{-}4)=\text{-}3(\text{-}3)(3)
f(-4)=9(3)f(\text{-}4)=9(3)
f(-4)=27f(\text{-}4)=27
The y-y\text{-}coordinate of the vertex is 27.27. Therefore, the vertex is the point (-4,27).(\text{-}4,27).

Draw the Parabola

Finally, we will draw the parabola through the vertex and the x-x\text{-}intercepts.