1998 Ford Taurus Car will not crank unless jumped.

Electrical problem
1998 Ford Taurus 6 cyl Two Wheel Drive Automatic

Hi. I have a 1998 Ford Taurus SE V6 3.0L, and I am having a confusing problem. Two weeks ago my car would not start all of a sudden when I came out of a store. It would make noise and try to turn over, but it just would not crank. I called a friend to come jump it, and without waiting longer than a minute, it cranked right up. I went a day later and bought a brand new battery to put in it. Then, today, two weeks later, I went to the gas station, put gas in the car, and when I went to start the car, it wouldn't crank again, yet it sounded like it was close. The same friend came out, jumped it, and it started in less than 30 seconds. It then drove about a mile to my house. I thought maybe it was the alternator, but I figured it would die on me while driving if it was the alternator. I also don't think it is the starter because it will only crank if jumped, though it make sounds and tries to turn over.