I was hoping someone could help me with this question, since I'm having trouble determining what approach to take.
Let $L \subseteq \{0,1\}^*$ be a regular language. Show the language $\{w \in \{0,1\}^* | 1w \in L\}$ is also regular.
My idea is that whenever we're at the start state to compensate for the lack of the 1, you do an additional transition as if 1 was part of the input and then carry on reading the input as normal.
Since L is regular it has a NFA that accepts it $(Q,\Sigma, \delta, q_0, F)$. Then construct a new NFA $(Q,\Sigma, \delta', q_0, F)$ such that $\delta'(q,a) = \delta(q,a)$ for $q \ne q_0$ and $\delta'(q,a) = \delta(\delta(q,a),1)$ for $q = q_0$
Is this correct?
This is almost correct. Note however that the given NFA may visit $q_0$ again, in which case you might drop additional 1s. Instead, what about $(Q,\Sigma,\delta, \delta(q_0,1),F)$?