Formally deduce that "$\forall p$ the vertex of $f_p$ will intersect the graph of $f$"

Question

Let $f_p(x)=x^2+2px$ and $f(x)=x^2-2x^2$.

Formally deduce i.e. construct a formal statement that "$\forall p$ the vertex of $f_p$ will intersect the graph of $f$".

Answer 1

The vertex of $f_p$ has $x$ coordinate given by $x_v = -p$ and $y$ coordinate $y_v = p^2 - 2p^2 = -p^2$. You just have to check if $f(x_v) = y_v$ or not. For function $f(x) = x^2 - 2x^2 = - x^2$ this is true.