Have to have Iphone 4 plugged up all the time to work?

I have an iphone 4. And if its not plugged up it won't last 5 mins. And when it dies, i will try to turn it back on and battery symbol shows all the way drained. All applications are closed. I tried that. Do you think i need another Battery?

No, leaving your iPhone plugged in all the time is not best for your iPhone 's battery long-term.
And the problem of drained battery too fast could be of many reason.

1. Its probably because you are using 3G and Wi-Fi connections or either accessive applications, that tends to drain the battery too fast.
2.Background Applications- If OpenSSH was installed on your iPhone, uninstall it using the installer app that came with the jailbreak software. In addition, turn off any push notifications by going to "Settings" and "Notifications." This background activity could be causing an unnecessary drain on your battery.
3. The iPhone's battery indicator may not be reading correctly. Test this by allowing your phone to run even if the battery indicator shows no power. If your phone stays on, you may need to download a battery control application and tweak the battery indicator sensitivity.

P.S: The battery life of your iPhone will naturally diminish over time. As you continue to use and recharge your device it will eventually stop holding a charge as long as it did when you first purchased it.