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Math Help - Algebra 2 help... Quadratic equations

  1. #1
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    Algebra 2 help... Quadratic equations

    9th grade here! I need help with a problem.. this is it:
    [Question]
    Write y=a(x-h)^2+k and y=a(x-p)(x-q) in standard form. Knowing the
    vertex of the graph of y=ax^2 + bx +c occurs at x=-b/2a, show that the vertex of the
    graph of y=a(x-h)^2+k occurs at x=h and that the vertex of the graph of y=a(x-p)(x-q)
    occurs at x=p+q/2.

    This is how far I got:
    y= a(x-h)^2+k
    >a(x^2-2xh-h^2)+k
    >y=ax^2-a2xh-ah^2+k
    >y=ax^2+(-2ah)x+(-ah^2=k)

    a=a b=-2ah c=ah^2+k

    >Next part:
    >a(x-p)(x-q)
    >a(x^2-qx-px+pq)
    >y=ax^2-aqx-apx+apq

    I'm stuck here, how to I arrange it to be standard form?
    Last edited by mathhelpplease; November 16th 2012 at 05:11 PM.
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  2. #2
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    Re: Algebra 2 help... Quadratic equations

    Hey mathhelpplease.

    It's been a while since I was in high school so can I ask what standard form is (for question 2)?
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  3. #3
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    Re: Algebra 2 help... Quadratic equations

    Hello, mathhelpplease!

    Your work seems to be correct,
    . . but I'm not sure where (or why) you are stuck.
    I'll walk through it for you.


    (a) Write y\,=\,a(x-h)^2+k and y\,=\,a(x-p)(x-q) in standard form.

    y \:=\:a(x-h)^2+k
    y\:=\:a(x^2-2hx + h^2) + k
    y\:=\:ax^2 - 2ahx + ah^2 + k
    y\:=\:ax^2 - 2ahx + (ah^2+k) .[1]

    y \:=\:a(x-p)(x-q)
    y\:=\:a(x^2 - px - qx + pq)
    y\:=\:ax^2 - apx - aqx + apq
    y\:=\:ax^2 - a(p+q)x + apq .[2]



    (b) Knowing the vertex of the graph of y=ax^2 + bx +c occurs at x=\tfrac{\text{-}b}{2a}
    show that the vertex of the graph of y\,=\,a(x-h)^2+k occurs at x=h
    and that the vertex of the graph of y\,=\,a(x-p)(x-q) occurs at x=\tfrac{p+q}{2.}

    [1] . y\:=\:ax^2 - 2ahx + (ah^2+k) \quad\Rightarrow\quad \begin{Bmatrix}a &=& a \\ b &=& \text{-}2ah \\ c &=& ah^2+k \end{Bmatrix}

    Vertex: . x \;=\;\frac{\text{-}b}{2a} \;=\;\frac{\text{-}(\text{-}2ah)}{2(a)} \;=\;\frac{2ah}{2a} \;=\;h

    \text{The vertex occurs at }\,x \,=\, h.


    [2] . y \:=\:ax^2 - a(p+q)x + apq \quad\Rightarrow\quad \begin{Bmatrix}a &=& a \\ b &=& \text{-}a(p+q) \\ c &=& apq \end{Bmatrix}

    Vertex: . x \;=\;\frac{\text{-}b}{2a} \;=\;\frac{\text{-}[\text{-}a(p+q)]}{2(a)} \;=\;\frac{a(p+q)}{2a} \;=\;\frac{p+q}{2}

    \text{The vertex occurs at }\,x \,=\,\tfrac{p+q}{2}
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  4. #4
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    Re: Algebra 2 help... Quadratic equations

    Quote Originally Posted by Soroban View Post
    Hello, mathhelpplease!

    Your work seems to be correct,
    . . but I'm not sure where (or why) you are stuck.
    I'll walk through it for you.



    y \:=\:a(x-h)^2+k
    y\:=\:a(x^2-2hx + h^2) + k
    y\:=\:ax^2 - 2ahx + ah^2 + k
    y\:=\:ax^2 - 2ahx + (ah^2+k) .[1]

    y \:=\:a(x-p)(x-q)
    y\:=\:a(x^2 - px - qx + pq)
    y\:=\:ax^2 - apx - aqx + apq
    y\:=\:ax^2 - a(p+q)x + apq .[2]




    [1] . y\:=\:ax^2 - 2ahx + (ah^2+k) \quad\Rightarrow\quad \begin{Bmatrix}a &=& a \\ b &=& \text{-}2ah \\ c &=& ah^2+k \end{Bmatrix}

    Vertex: . x \;=\;\frac{\text{-}b}{2a} \;=\;\frac{\text{-}(\text{-}2ah)}{2(a)} \;=\;\frac{2ah}{2a} \;=\;h

    \text{The vertex occurs at }\,x \,=\, h.


    [2] . y \:=\:ax^2 - a(p+q)x + apq \quad\Rightarrow\quad \begin{Bmatrix}a &=& a \\ b &=& \text{-}a(p+q) \\ c &=& apq \end{Bmatrix}

    Vertex: . x \;=\;\frac{\text{-}b}{2a} \;=\;\frac{\text{-}[\text{-}a(p+q)]}{2(a)} \;=\;\frac{a(p+q)}{2a} \;=\;\frac{p+q}{2}

    \text{The vertex occurs at }\,x \,=\,\tfrac{p+q}{2}
    Thank you so much. That cleared things up but I wanted to know how you went from y=ax^2-apx-aqx+apq to y=ax^-a(p+q)x+apq
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