Locus of The Vertex

**JohnDoe** · February 14th 2008, 11:34 AM

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

**earboth** · February 14th 2008, 11:58 AM

Originally Posted by JohnDoe

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)

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