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Math Help - Writing the vertex of a parabola

  1. #1
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    Writing the vertex of a parabola

    Just like a second opinion please.

    I have a parabola y = x^2 - 10x - 8

    I have the equation y = a(x - h)^2 + K

    where (h, k) are the vertex

    am I right to change the signs and write;

    h = 10
    k = 8

    so the vertex of y = x^2 - 10x - 8

    = (10, 8)

    Thanks
    Last edited by David Green; August 10th 2011 at 12:27 PM. Reason: put the equation in place of the parabola mix up
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Writing the vertex of a parabola

    You've to convert y=x^2-10x-8 into the form y=a(x-h)^2+k.
    In you other topic you used the technique 'completing the square', use this technique for this exercice.

    Notice:
    The vertex V(10,8) is the vertex of the parabola: y=(x-10)^2+8=x^2-20x+108.
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  3. #3
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    Re: Writing the vertex of a parabola

    Quote Originally Posted by Siron View Post
    You've to convert y=x^2-10x-8 into the form y=a(x-h)^2+k.
    In you other topic you used the technique 'completing the square', use this technique for this exercice.

    Notice:
    The vertex V(10,8) is the vertex of the parabola: y=(x-10)^2+8=x^2-20x+108.
    Thanks for that, so if I already have the equation and the parabola, I can just read straight from the parabola and apply it to the h and k, i.e the vertex (10, 8).
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  4. #4
    MHF Contributor Siron's Avatar
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    Re: Writing the vertex of a parabola

    If you've a polynomial in the form:
    y=a(x-h)^2+k
    then like you said, you can determine the coordinates of the vertex directly: (h,k)

    If you've a polynomial in the form:
    y=ax^2+bx+c
    then you've to convert this polynomial into the form y=a(x-h)^2+k

    In case of your exercice, you have given a polynomial in the form y=ax^2+bx+c so convert it to the other form (by using 'completing the square, like you did in your previous topic)

    So, can you write:
    y=x^2-10x-8 into the form: y=a(x-h)^2+k?
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  5. #5
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    Re: Writing the vertex of a parabola

    Quote Originally Posted by Siron View Post
    If you've a polynomial in the form:
    y=a(x-h)^2+k
    then like you said, you can determine the coordinates of the vertex directly: (h,k)

    If you've a polynomial in the form:
    y=ax^2+bx+c
    then you've to convert this polynomial into the form y=a(x-h)^2+k

    In case of your exercice, you have given a polynomial in the form y=ax^2+bx+c so convert it to the other form (by using 'completing the square, like you did in your previous topic)

    So, can you write:
    y=x^2-10x-8 into the form: y=a(x-h)^2+k?
    If I am right, then;

    y = x^2 - 10x + 8 = x(x - 10)^2 - 8
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  6. #6
    MHF Contributor Siron's Avatar
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    Re: Writing the vertex of a parabola

    But y=x(x-10)^2-8=x(x^2-20x+100)-8=x^3-20x^2+100x-8, so that's not an option.
    Do you notice:
    x^2-10x+25=(x-5)^2
    ?
    But in this case you've y=x^2-10x-8, so you've to subtract a (constant) number of 25 to hold the original equation, so:
    x^2-10x+25-a=x^2-10x-8

    Calculate a, then you can wright:
    x^2-10x-8=(x-5)^2-a
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  7. #7
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    Re: Writing the vertex of a parabola

    Quote Originally Posted by David Green View Post
    I have a parabola y = x^2 - 10x - 8
    I have the equation y = a(x - h)^2 + K
    where (h, k) are the vertex
    y=x^2-10x-8=(x-5)^2-33 Completing the square.
    So h=5~\&~k=-33.
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  8. #8
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    Re: Writing the vertex of a parabola

    Quote Originally Posted by Siron View Post
    But y=x(x-10)^2-8=x(x^2-20x+100)-8=x^3-20x^2+100x-8, so that's not an option.
    Do you notice:
    x^2-10x+25=(x-5)^2
    ?
    But in this case you've y=x^2-10x-8, so you've to subtract a (constant) number of 25 to hold the original equation, so:
    x^2-10x+25-a=x^2-10x-8

    Calculate a, then you can wright:
    x^2-10x-8=(x-5)^2-a
    Thanks Siron, Plato has just beat me to it with the solution, but I have worked it out like this;

    y = x^2 - 10x - 8 = (x - 5)^2 - 25 - 8

    = (x - 5)^2 - 33

    vertex is (5, - 33)

    Thanks to you both much appreciated.
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  9. #9
    MHF Contributor Siron's Avatar
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    Re: Writing the vertex of a parabola

    Well done and you're welcome!
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  10. #10
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    Re: Writing the vertex of a parabola

    Quote Originally Posted by Siron View Post
    Well done and you're welcome!
    Thanks

    David
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