Listen to these people and get an instructor to take you into some of the higher airports then you will understand.
I took off in a 172 at DA 7000' ish. I checked the books , did the sums, didn't beleive it was going to take 5000' of the 9000' to reach rotate speed but it did and then climbed at 150fpm. On another trip I took off from sandiego to go to Little bear (the one at 6200'). The plane was so tired it barely made 400fpm at SD so I scrubbed the flight and went up thecoast instead. I doubt it would have ever reached rorate speed at 6200'.
If you are going to extrapolate be sure of your capabilities. Its challenging work flying up there but the views are great. Its an experience I remember frequently.