Here are some guiding principles:
* Keep it concise: Short introductory phrases are generally easier to read. Aim for 4-5 words or less.
* Avoid ambiguity: A comma should be used after an introductory phrase to clearly separate it from the main clause. If the phrase is too long, it might create confusion.
* Consider the sentence structure: If the introductory phrase is complex or contains multiple clauses, it might be better to break it into separate sentences.
Examples:
Short and clear:
* After a long day, I was ready for bed.
* Despite the rain, the game continued.
Longer but still clear:
* Having finished her homework, she went out to play.
* To make a delicious cake, you need to follow the recipe carefully.
Potentially confusing:
* After having spent the entire day cleaning the house and doing laundry, I was exhausted. (This could be clearer by breaking it into two sentences.)
Ultimately, the best way to determine if your introductory phrase needs a comma is to read the sentence aloud and see if it flows naturally. If it sounds awkward or confusing, you may need to revise it.