The authors study the behavior at infinity of the Bergman kernel function and the Bergman metric on unbounded model domains in \(\mathbb{C}^{n+1}\) of the form \(\{\,(z,w): \Im w > \rho(z)\,\}\), where \(\rho\)~is a non-negative plurisubharmonic function on~\(\mathbb{C}^n\) that tends to infinity with~\(\| z\| \). They show first of all that the Bergman kernel function is non-trivial and the Bergman metric is complete. Moreover, in any tubular neighborhood of the complex line \(z=0\) in~\(\mathbb{C}^{n+1}\), the Bergman kernel on the diagonal at \((z,w)\) tends to zero faster than \((\Im w)^{-2}\) as \(\Im w \to \infty\). Additionally, the authors obtain precise estimates on the size at infinity of the Bergman kernel and Bergman metric under various special hypotheses on the growth of~\(\rho\).