jmc

Since the vertex is $(p,p),$ the parabola is of the form $$y=a(x-p{)}^2+p.$$Setting $x=0,$ we get $y=a{p}^2+p=-p,$ so $a=-\frac{2}{p}.$ Then $$y=-\frac{2}{p}(x^2-2px+p^2)+p=-\frac{2}{p}{x}^2+4x-p,$$so $b=\boxed{4}.$