We analyze the boundedness of the wavelet transform ${\mathcal W}_g f$of the quasiasymptotically bounded distribution $f$.Assuming that the distribution $f\in\mathcal{S}'(\mathbb R)$is quasiasymptotically or $r$-quasiasymptotically bounded at a point or at infinityrelated to a continuous and positive function,we obtain results for the localization of its wavelet transform.