would a bad alternator cause the check engine light come on?

Hello, I'm Danny.

Yes, a failing or defective alternator can cause a check engine light. Most auto parts stores will test your alternator for free. Here are two tutorials showing how to test and replace an alternator:

https://www.2carpros.com/articles/how-to-check-a-car-alternator

https://www.2carpros.com/articles/how-to-replace-an-alternator

I would also suggest checking the codes that might be stored in your cars computer from the check engine light being on. If you don't have access to a scanner or code reader most auto parts stores have a tool loaner program and can rent you one. Here is a tutorial showing how to check for codes:

https://www.2carpros.com/articles/checking-a-service-engine-soon-or-check-engine-light-on-or-flashing

After you perform these tests get back to us with the information and we can then properly diagnose these issues. Hope this helps and thanks for using 2CarPros.