In this paper we consider a large class of super-Brownian motions in $\mathbb{R}$ with spatially dependent branching mechanisms. We establish the almost sure growth rate of the mass located outside a time-dependent interval $(-δt,δt)$ for $δ>0$. The growth rate is given in terms of the principal eigenvalue $λ_{1}$ of the Schödinger type operator associated with the branching mechanism. From this result we see the existence of phase transition for the growth order at $δ=\sqrt{λ_{1}/2}$. We further show that the super-Brownian motion shifted by $\sqrt{λ_{1}/2}\,t$ converges in distribution to a random measure with random density mixed by a martingale limit.