H(t)=vt-5t2=-5(t2-vt/5)=-5(t2-vt/5+v2/100-v2/100),
H(t)=-5((t-v/10)2-v2/100)=-5(t-v/10)2+v2/20. This has maximum value when time t=v/10, so the maximum height H(v/10)=v2/20.
If x=H(t), then we can write this as x-v2/20=-5(t-v/10)2 and the vertex is (t,x)=(v/10,v2/20). (Vertex form is usually written y-k=a(x-h)2, with vertex at (h,k), so here we replace y with x and x with t. k=v2/20 and h=v/10.)