The authors present some oscillation and nonoscillation theorems for the following nonlinear delay difference equation: \[ x(n+1) -ax(n) +f(n,x(n-k)) =0,\ n\in\mathbb{Z}^{+}:=\left\{ 0,1,2,\dots\right\} . \] Here, \(a\) is a positive number, \(k\) is a positive integer, and \(f:\mathbb{Z}^{+}\times\mathbb{R}\rightarrow\mathbb{R}\) is a continuous function with respect to the second variable that satisfies \(f(n,0) =0\) for \(n\in\mathbb{Z}^{+}\) and \[ xf(n,x) >0\text{ for }n\in\mathbb{Z}^{+},\ x\neq0. \] The obtained results are proved by means of phase plane analysis for a system equivalent to the above equation. The results extend theorems in the literature.