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:
$\displaystyle f(x) = x^2 - 2mx + 2m +1~\iff~f(x) = x^2-2mx + m^2 - m^2 + 2m +1 $ $\displaystyle = (x-m)^2-m^2+2m+1$ . Therefore the coordinates of the vertex are:
V(m, -m^2+2m+1)
2. From the coordinates you know:
$\displaystyle \left|\begin{array}{l}x = m\\ y = -m^2 + 2m +1\end{array}\right.$
3. Substitute the term for m from the 1rst equation into the 2nd equation. You get:
$\displaystyle y = -x^2+2x+1$ . And that's the equation of the curve on which all vertices of all parabola are placed. (Red line)