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Math Help - Locus of The Vertex

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    Post Locus of The Vertex

    Hi, I got a question asking "Find the equation of the locus of the vertex of f(x)=x^2-2mx+2m+1 "What is locus is and how can i write an equation of a locus i made a search but didnt really get anything useful. Thanks
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  2. #2
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    Quote Originally Posted by JohnDoe View Post
    Hi, I got a question asking "Find the equation of the locus of the vertex of f(x)=x^2-2mx+2m+1 "What is locus is and how can i write an equation of a locus i made a search but didnt really get anything useful. Thanks
    1. Calculate the coordinates of the vertex:

    f(x) = x^2 - 2mx + 2m +1~\iff~f(x) = x^2-2mx + m^2 - m^2 + 2m +1  = (x-m)^2-m^2+2m+1 . Therefore the coordinates of the vertex are:

    V(m, -m^2+2m+1)

    2. From the coordinates you know:

    \left|\begin{array}{l}x = m\\ y = -m^2 + 2m +1\end{array}\right.

    3. Substitute the term for m from the 1rst equation into the 2nd equation. You get:

    y = -x^2+2x+1 . And that's the equation of the curve on which all vertices of all parabola are placed. (Red line)
    Attached Thumbnails Attached Thumbnails Locus of The Vertex-vertexlocus.gif  
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