In a model category $C$ and $X \in C$ we can define $\Sigma X$ as the homotopy pushout of $$* \leftarrow X \rightarrow * $$ letting $*$ denote the terminal object.
The way I understand it is that if $C = \text{Ch}_+(R-\text{Mod})$ with the projective model structure, if $D$ is a chain complex, $\Sigma D = D[-1]$ where $D[-1]_n = D_{n-1}$. I believe this is true because of the case when $D = C_*(X;R)$, where $X$ is some topological space.
Consider the map
$$C_*(X;R) \rightarrow C_*(C X;R)$$
where $CX$ is the cone over $X$ and note that it is a cofibrant replacement for $C_*(X) \rightarrow *$ in this case and we can thus deduce from some basic algebraic topology that $\Sigma C_*(X;R) \approx C_*(\Sigma X;R)$ which is weakly equivalent as a chain complex to $C_*(X;R)[-1]$.
First of all, is my intuition for this correct? Is $\Sigma D = D[-1]$ for general $D$? I have no clue how to generalize this argument to general chain complexes $D$ so any help would be greatly appreciated.