From Spivak's Physics for Mathematicians, vol.1:
At this point, it might be nice to have a few concrete exzamples of canonical transfomrations $f:T^{*}M\to T^{*}M$. Canonical transformations are often defined in terms of the corresponding change of variables: given coordinates $(q,p)$ on the domain, we give a formula for the corresponding coordinates $$ \begin{matrix} Q^{i}=q^{i}\circ f \\ P_j = p_j\circ f \end{matrix} \qquad \text{with}\qquad \begin{matrix} dQ^{i}=f^{*}(dq^{i})\\ dP_j=f^{*}(dp_j), \end{matrix} $$ on the range. For example, in dimension $1$, we can consider $$ Q=q+p\tau +\frac{1}{2}g\tau ^2,\qquad P=p+g\tau $$ for constants $g$ and $\tau$. Since $$ dQ\wedge dP=f^{*}(dq+\tau dp)\wedge f^{*}dp=f^{*}(dq\wedge dp), $$ this is canonical transformation. Often, we don't even bother writing the $f^{*}$.
In the last string of equalities, shouldn’t $ f^*(dq + \tau dp)\wedge f^*dp $ instead read $(dq + \tau dp)\wedge dp$?
$Q, P, q, p$ are coordinate functions on $T^*M$, and $f:T^*M \rightarrow T^*M$. If we have $Q=q+\tau p+\frac{1}{2}gt^2$ and $P=p+\tau g$, then $dQ=dq+\tau dp$ and $dP=dp$. So $dQ\wedge dP=(dq+\tau dp)\wedge dp$. On the other hand, since $Q=f^*q$ and $P=f^*p$, we have $dQ=f^*dq$ and $dP=f^*dp$. So $f^*dq=dq+\tau dp$, $f^*dp=dp$, and
