This article investigates the approximation properties of a general family of positive linear operators defined on the unbounded interval [0, ∞). We prove uniform convergence theorem and Voronovskaya-type theorem for functions with polynomial growth. More precisely, we study weighted approximation i.e basic convergence theorems, quantitative Voronovskaya-asymptotic theorems and Grüss Voronovskaya-type theorems in weighted spaces. Finally, we obtain the rate of convergence of these operators via a suitable weighted modulus of continuity.