Final answer:
The correct punctuation for Julie's first draft response is "offered a determined response; her enthusiasm," as it appropriately uses a semicolon to join two related independent clauses.
Step-by-step explanation:
In Julie's first draft response that touches upon the theme of parental influence on children’s lives, the correct punctuation for the sentence provided should be "Cathy’s face: b) offered a determined response; her enthusiasm." This choice uses a semicolon to correctly join two related independent clauses. The use of the semicolon indicates that the enthusiasm is a consequence or a significant addition to Cathy’s determined response. This serves both to avoid a comma splice and to maintain the flow of the sentence, making it clear and easy to follow.