time machine without the original disc?

Time Machine will back up your entire system by default. Once completed, you can completely erase your MacBook Pro and restore it from the backup.

The most obvious application for that would be the need to replace your MacBook Pro's internal disk drive, or, if you were to experiment with it for whatever reason and desired to revert to the state that existed prior to your experimentation.

I'm not sure what you mean by "the original disc" though. What disc are you referring to?

Yes, if you bought your Mac "used" then you should absolutely erase it completely and reinstall its original operating system from its original discs, unless it shipped with Lion or Mountain Lion already installed. That's the best way to ensure complete eradication of whatever junk may be responsible for its slow performance, and it's also required to legally transfer its OS X license to you.

If your Mac was built prior to (approximately) August, 2011 and you don't have the discs, purchase a replacement set from Apple by contacting them by phone using the number in the Apple Online Store. You will need the Mac's serial number and payment for a nominal fee ($25 or so). Explain that you need to purchase a replacement System Install disc for your Mac. Use those words so they don't misunderstand what you require.

Any Mac that shipped with discs must keep those discs from cradle to grave, so when Apple sends you a replacement don't lose it. Maintenance such as you describe can be a challenge without them.

New Macs don't ship with discs and don't need them. If it shipped with Lion or later OS already installed, you can completely erase its disk and reinstall OS X using OS X Internet Recovery. No Apple ID is required to do that.

Though you should back up your system before erasing it, backing up using Time Machine and subsequently restoring it from that backup won't change a thing about its performance.