$E(Y)=E(\alpha x+\beta )=E(\alpha x)+E(\beta )=\alpha E(x)+\beta =\alpha \mu +\beta$
Since E(ax)=aE(x) and E(constant)=constant
$Var(Y)=Var(\alpha x+\beta )=Var(\alpha x)+Var(\beta )=\alpha^{2} Var(x)+\beta =\alpha^{2} \sigma ^{2}$
Since Var(ax)=$a^{2}$Var(x) and Var(constant)=0
Hence, Y is a N($\alpha \mu +\beta$, $\alpha^{2} \sigma ^{2}$) random variable