Slowly failing alternator?

I think my alternator is starting to go bad, but I wanted to get some confirmation. Here's the deal:

The engine was frequently slow to crank when starting, especially after lenghty drives. The weird thing was that it wasn't "slowly cranking", it was either not turning over or cranking at what seemed like full speed. I could turn the key and . . . nothing. Try again . . . nothing. But if I held it at start for a few seconds, it would finally turn over and start right up. I thought maybe it was a wiring thing, except for the eventual cranking when held at start. Anyway, it eventually wouldn't turn over at all. Suspecting my old battery, I got a jump (started right up), promptly went to the auto parts store, and replaced it. A few days later, I again noticed that it was slow to turn over after long drives. Sounds like alternator, right?